#7330694

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,41)(20,24,31,38)(21,33,34,25)(23,35,37,30)(28,40,43,48)(32,45,44,39)(36,46,47,42);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,19,21,30,22,29,34,35)(18,23,25,27,26,37,33,41)(20,28,32,42,31,43,44,46)(24,36,39,40,38,47,45,48);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20,22,31)(18,24,26,38)(19,28,29,43)(21,32,34,44)(23,36,37,47)(25,39,33,45)(27,40,41,48)(30,42,35,46);(2,6)(5,7)(10,11)(14,15)(19,29)(20,31)(23,37)(24,38)(27,41)(30,35)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,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, 7237890);