#469627

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,9)(7,11)(21,22,27,35,26,31,25,30)(23,28,32,36,24,34,33,29);(5,6,8,9)(7,12,11,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,26)(25,33)(27,32)(29,30)(31,34)(35,36);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,26,23)(22,29,31,36)(25,33,27,32)(28,35,34,30);(2,4)(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,27)(22,30,31,35)(23,32,24,33)(28,36,34,29)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=i,d^2=eg,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, 376823);