#246562

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153758);