#303036

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

Aliases

No aliases for this group.

Subgroups

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

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=f,d^2=eh,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, 210232);