#7230811

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 7138007);