#454997

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1428 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)(9,10,12,13)(11,14,15,16)(17,18,21,25,22,20,23,26)(19,27,30,31,29,24,28,32);(1,2)(3,4)(10,13)(14,16)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31);(2,4)(11,15)(14,16)(17,20,22,18)(19,24,29,27)(21,25,23,26)(28,32,30,31);(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,20,26)(19,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,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, 362193);