#238742

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1122 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,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, 145938);