#279022

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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)(7,12)(8,17)(9,19)(10,14)(11,16)(13,18)(15,20)(21,22,26,28,24,29,25,30)(23,31,34,36,32,27,33,35);(1,2)(3,4)(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20)(21,23,24,32)(22,27,29,31)(25,33,26,34)(28,35,30,36);(2,4)(7,15)(8,16)(11,17)(12,20)(22,28)(25,26)(27,35)(29,30)(31,36)(33,34);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)(22,29)(23,32)(28,30)(33,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186218);