#243072

order	512
nilpotency class	3
isoclinism	isoclinic class of #243068
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150268);