#238744

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1046 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 145940);