#233307

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
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×#3184	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=eh,f^c=fi,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, 140503);