#472172

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 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)(5,6)(7,9)(8,10)(11,12)(14,16)(18,19)(21,22,25,28,26,30,24,29)(23,27,33,31,34,36,32,35);(5,7,8,11)(6,9,10,12)(13,14,17,16)(15,19,20,18)(21,23,25,33,26,34,24,32)(22,27,28,31,30,36,29,35);(2,4)(13,15,17,20)(14,18,16,19)(21,22)(23,31)(24,28)(25,29)(26,30)(27,33)(32,36)(34,35);(6,10)(9,12)(13,16,17,14)(15,18,20,19)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,31,36,35)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379368);