#7511106

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1271 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,28)(20,33)(21,37)(22,25)(23,27)(24,32)(26,36)(29,39)(30,40)(31,41)(34,42)(35,43)(38,44)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(17,19,22,32)(18,24,25,28)(20,34,36,30)(21,31,23,29)(26,42,33,40)(27,41,37,39)(35,46,38,45)(43,48,44,47);(10,13)(14,15)(18,25)(19,29)(21,23)(24,39)(26,33)(28,41)(30,45)(31,32)(34,46)(35,38)(40,47)(42,48);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,20,22,36)(18,26,25,33)(19,30,32,34)(21,38,23,35)(24,40,28,42)(27,44,37,43)(29,45,31,46)(39,47,41,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,37)(19,31,32,29)(20,35,36,38)(24,41,28,39)(26,43,33,44)(30,45,34,46)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7418302);