#416361

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,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, 323557);