#410402

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=gi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,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, 317598);