#323834

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231030);