#234656

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dh,e^c=ei

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, 141852);