#454854

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

1298 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)(10,13)(11,15)(17,18,23,31,21,26,22,27)(19,28,30,32,20,24,29,25);(1,2)(3,4)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,24,26,28)(22,29,23,30)(25,27,32,31);(2,4)(9,11)(10,14)(12,15)(13,16)(17,20,21,19)(18,25,26,32)(22,29,23,30)(24,27,28,31);(6,8)(11,15)(14,16)(19,20)(24,28)(25,32)(29,30)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,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, 362050);