#470158

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

846 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)(10,11)(13,14,17,16)(15,18,19,20)(23,30)(24,25)(27,35)(28,34)(31,36)(32,33);(5,6,8,9)(7,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,22,26,29)(23,31,33,27)(24,34,25,28)(30,36,32,35);(5,7)(6,10)(8,12)(9,11)(15,19)(18,20)(21,23,25,30,26,33,24,32)(22,27,28,35,29,31,34,36);(2,4)(7,12)(10,11)(13,16,17,14)(15,20,19,18)(21,24,26,25)(22,28,29,34)(23,32,33,30)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=egi,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, 377354);