#247118

order	512
nilpotency class	3
isoclinism	isoclinic class of #247047
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1217 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^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,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154314);