#304162

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=efh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211358);