#455266

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362462);