#432357

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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)(10,12)(13,14,16,17)(15,18,19,20)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(5,6,8,9)(7,10,11,12)(14,17)(15,19)(21,23,24,30)(22,26,27,34)(25,29,32,31)(28,36,35,33);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(2,4)(6,9)(10,12)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=efi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,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, 339553);