#246773

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1205 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=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, 153969);