#219488

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,7,5,3,6,8)(9,10,15,19,16,21,14,20)(11,22,24,27,25,32,17,12)(13,28,30,35,31,34,29,18)(23,37,38,40,39,26,33,36);(1,3,4,8,5,2,6,7)(9,11)(10,12,21,27)(13,23)(14,17)(15,24)(16,25)(18,26,35,37)(19,22,20,32)(28,36,34,40)(29,33)(30,38)(31,39);(9,12)(10,17,21,24)(11,19,25,20)(13,26)(14,22)(15,32)(16,27)(18,33,35,38)(23,28,39,34)(29,36)(30,40)(31,37);(9,13,15,30,16,31,14,29)(10,18,20,34,21,35,19,28)(11,23,24,38,25,39,17,33)(12,26,32,40,27,37,22,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126684);