#234803

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3217	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1601 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)(7,12)(8,13)(9,11)(10,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(24,31)(26,33)(27,35)(29,32);(2,4)(5,7,10,16)(6,12,14,20)(8,17,9,15)(11,18,13,19)(21,23)(22,26)(24,32)(25,30)(27,36)(28,33)(29,31)(34,35);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(21,24,25,31)(22,27,28,35)(23,29,30,32)(26,34,33,36)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,d^c=dhi,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, 141999);