#7233791

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7140987);