#469801

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

870 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)(13,14,17,18)(15,19,20,16)(22,26)(23,29)(25,32)(27,34)(30,35)(31,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,16,18,19)(21,22)(23,30)(24,26)(25,28)(27,31)(29,34)(32,33)(35,36);(6,9)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23,25,29,24,31,32,36)(22,27,28,35,26,30,33,34);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(16,19)(22,26)(23,31)(28,33)(29,36)

Polycyclic presentation:

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

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, 376997);