#7510249

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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)(9,10,13,15)(11,14,12,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,45)(31,40)(33,39)(34,48)(35,42)(36,43)(37,44)(38,46)(41,47);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,33)(22,32)(23,31)(25,41)(26,40)(27,29)(28,39)(30,35)(36,47)(37,46)(38,42)(43,48)(44,45);(9,12,13,11)(10,14,15,16)(17,20,22,36)(18,25,27,43)(19,30,32,46)(21,35,23,37)(24,38,29,45)(26,42,28,44)(31,34,33,47)(39,41,40,48);(2,7)(3,8)(10,15)(14,16)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,40)(25,42,43,44)(30,34,46,47)(38,41,45,48);(1,4,5,6)(2,3,7,8)(18,27)(19,32)(25,43)(26,28)(30,46)(31,33)(34,47)(42,44)

Polycyclic presentation:

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

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, 7417445);