#305402

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,d^b=di,e^b=eh,g^b=gi,g^c=gi,e^d=eh

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, 212598);