#94825

order	512
nilpotency class	7
isoclinism	isoclinic class of #94823
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7529
commutator	C₈²	C₂×C₄
Frattini, cosocle	#1713	C₂²
prime radical	#1712	C₄
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1713 ◁ G
LECS	G ▷ #1713 ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fg,d^2=i,e^2=gh,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cfgh,d^b=def,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2021);