#135833

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1853
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8830
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225416, #72225466, #72225467
maximal normal subgroup of	#72225416, #72225466, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43029);