#138657

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1667
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9221
prime radical	#33594	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,14,8,15,32,33,9,16,31,44,7,12,5,13)(3,10,20,36,21,38,19,35,22,39,29,45,18,37,53,61)(4,23,26,43,27,46,25,41,28,56,30,11,24,40,57,42)(17,47,49,34,51,55,54,59,52,62,64,60,50,63,48,58);(1,3,6,19,8,18,32,20,9,22,31,53,7,21,5,29)(2,10,33,36,12,38,14,35,16,39,13,45,15,37,44,61)(4,17,25,49,24,51,26,54,28,52,57,64,27,50,30,48)(11,34,40,62,42,58,23,55,43,60,46,47,41,59,56,63);(1,4)(2,11)(3,17)(5,30)(6,25)(7,27)(8,24)(9,28)(10,34)(12,42)(13,46)(14,23)(15,41)(16,43)(18,51)(19,49)(20,54)(21,50)(22,52)(26,32)(29,48)(31,57)(33,40)(35,55)(36,62)(37,59)(38,58)(39,60)(44,56)(45,47)(53,64)(61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225389, #72225396, #72225399
maximal normal subgroup of	#72225389, #72225396, #72225399
maximal quotient of	#72225511

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45853);