#463548

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
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

964 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ei,c^2=efi,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370744);