#94748

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	D₈
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	#7458
prime radical	C₂²	#1422
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

57 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ce,c^2=ei,d^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225395, #72225397, #72225398, …
maximal normal subgroup of	#72225395, #72225397, #72225398, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1944);