#234802

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	#37546	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1515 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)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(5,7,10,17)(6,11,13,19)(8,16,9,15)(12,20,18,14)(24,31)(26,33)(27,35)(29,32);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19)(21,23)(22,26)(24,32)(25,30)(27,36)(28,33)(29,31)(34,35);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(21,24,25,31)(22,27,28,35)(23,29,30,32)(26,34,33,36)

Polycyclic presentation:

2^9|b^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, 141998);