#92864

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1270
prime radical	#3147	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

206 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,9,5,7,11,14,6,10,13,16,4,8,12,15)(18,22,23,30,25,31,24,26);(2,7)(3,11)(4,5)(8,10)(9,16)(12,13)(17,18)(19,26)(20,23)(21,25)(22,28)(24,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72210667, #72210670, #72210672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60);