#338672

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245868);