#7248066

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42477	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7155262);