#7230471

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,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, 7137667);