#233666

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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	C₂×#3180	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 140862);