#7330825

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(17,18,22,27)(19,28,29,43)(20,24,34,40)(21,25,30,26)(23,32,38,33)(31,45,35,48)(36,42,37,44)(39,47,41,46);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(19,29)(20,34)(23,38)(24,40)(28,43)(32,33)(36,37)(42,44);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,19,30,32,22,29,21,33)(18,23,25,43,27,38,26,28)(20,35,36,46,34,31,37,47)(24,41,44,45,40,39,42,48);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,31)(21,37)(22,34)(23,39)(25,44)(26,42)(27,40)(28,45)(29,35)(30,36)(32,47)(33,46)(38,41)(43,48);(2,6)(5,7)(10,11)(14,15)(17,21,22,30)(18,25,27,26)(19,32,29,33)(20,36,34,37)(23,28,38,43)(24,42,40,44)(31,46,35,47)(39,48,41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7238021);