#399495

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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)(5,6,9,10)(7,11,8,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,30)(23,27,33,36)(24,28,25,29)(31,34,32,35);(6,10)(11,12)(13,15,17,16)(14,18,19,20)(21,23)(22,27)(24,32)(25,31)(26,33)(28,35)(29,34)(30,36);(5,7,9,8)(6,11,10,12)(13,16,17,15)(14,18,19,20)(23,31)(24,25)(27,34)(28,29)(32,33)(35,36);(2,4)(5,8,9,7)(6,12,10,11)(21,24,26,25)(22,28,30,29)(23,32,33,31)(27,35,36,34)

Polycyclic presentation:

2^9|a^2=ei,b^2=f,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,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, 306691);