#7331077

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,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=dhi,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, 7238273);