#402992

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1452 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22,25,29,27,31,26,30)(23,32,33,36,24,35,34,28);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,27,24)(22,28,31,36)(25,34,26,33)(29,35,30,32);(15,19)(17,20)(21,24,27,23)(22,28,31,36)(25,33,26,34)(29,35,30,32);(2,4)(5,8,9,7)(6,10,11,12)

Polycyclic presentation:

2^9|a^2=eg,b^2=efi,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310188);