#239314

order	512
nilpotency class	3
isoclinism	isoclinic class of #239311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60734	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,24)(13,18)(14,29)(15,19)(16,22)(17,27)(21,38)(23,32)(25,37)(26,36)(28,35)(30,33)(31,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,32)(19,33)(20,29)(21,27)(24,31)(26,40)(28,39)(35,38)(36,37);(5,8)(9,12,14,26,15,27,13,28)(10,17,18,36,19,24,29,35)(11,21,22,40,23,25,30,39)(16,31,20,37,33,34,32,38);(9,13,15,14)(10,18,19,29)(11,22,23,30)(12,26,27,28)(16,32,33,20)(17,35,24,36)(21,39,25,40)(31,37,34,38)

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,f^c=fh,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, 146510);