#446486

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

1108 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)(6,10)(11,12)(21,22,25,29)(23,26,31,35)(24,27,32,28)(30,33,36,34);(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,20,17)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(2,4)(5,8,9,7)(6,11,10,12)

Polycyclic presentation:

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

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, 353682);