#93868

order	512
nilpotency class	5
isoclinism	isoclinic class of #93866
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7096
commutator	C₈²	C₂×C₄
Frattini, cosocle	C₈×C₁₆	C₂²
prime radical	#1712	C₄
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,c^2=fh,d^2=efhi,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fh,g^b=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214160, #72214214, #72214234, …
maximal normal subgroup of	#72214160, #72214214, #72214234, …
maximal quotient of	#72216560, #72216565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1064);