#304112

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 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)(9,10,12,14)(11,16,15,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,5,4)(2,6,8,7)(10,14)(13,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(1,3,5,4)(2,7,8,6)(9,11,13,10,12,15,16,14)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,4,5,3)(2,7,8,6)(10,14)(11,15)(18,23)(22,25)(24,31)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=efh,d^2=ehi,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=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211308);