#450844

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

922 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)(5,6,9,8)(7,11,12,10)(13,14,17,19)(15,18,16,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15,17,16)(14,18,19,20)(21,23,24,31)(22,26,27,29)(25,30,33,32)(28,34,36,35);(5,8,9,6)(7,11,12,10)(14,19)(18,20)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(2,4)(7,12)(10,11)(13,16,17,15)(14,20,19,18)(22,27)(23,31)(28,36)(30,32)

Polycyclic presentation:

2^9|a^2=efh,b^2=fi,c^2=ei,d^2=f,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, 358040);