#472438

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=f,d^2=ei,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, 379634);