#231037

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138233);