#234804

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×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1385 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)(7,14)(8,11)(9,13)(10,19)(12,17)(15,16)(18,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(5,7,11,18)(6,12,8,15)(9,17,10,16)(13,20,19,14)(24,31)(26,33)(27,35)(29,32);(2,4)(5,8,11,6)(7,15,18,12)(9,13,10,19)(14,17,20,16)(21,23)(22,26)(24,32)(25,30)(27,36)(28,33)(29,31)(34,35);(5,9,11,10)(6,13,8,19)(7,16,18,17)(12,14,15,20)(21,24,25,31)(22,27,28,35)(23,29,30,32)(26,34,33,36)

Polycyclic presentation:

2^9|b^2=h,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, 142000);