#233128

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3100	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1444 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)(7,15)(8,14)(9,13)(10,12)(11,16)(17,19)(18,20)(21,22,26,30)(23,31,33,36)(24,29,25,28)(27,34,35,32);(2,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,20)(14,15);(7,16)(8,9)(11,15)(13,14)(17,18)(19,20)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,36)(30,35);(6,12)(11,19)(13,14)(15,20)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,31,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140324);