#233672

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
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	C₂×C₄²	C₂⁴
prime radical	#37606	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1373 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140868);