#232387

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	#37556	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1353 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)(7,16)(8,13)(9,20)(10,15)(11,14)(12,18)(17,19)(22,27)(24,25)(28,31)(29,30)(32,33)(35,36);(5,7)(6,12)(8,14)(9,13)(10,18)(11,17)(15,19)(16,20)(21,22,26,30)(23,31,34,35)(24,29,25,27)(28,33,36,32);(2,4)(5,8)(6,13)(7,15)(9,19)(10,16)(11,12)(14,18)(17,20)(21,23)(22,28)(24,32)(25,33)(26,34)(27,31)(29,35)(30,36);(6,14)(7,17)(13,18)(15,20)(21,24,26,25)(22,29,30,27)(23,32,34,33)(28,35,36,31)

Polycyclic presentation:

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

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, 139583);