#151904

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12611
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=di,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225380, #72225566, #72225568, …
maximal normal subgroup of	#72225380, #72225566, #72225568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59100);