#7600509

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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)(3,7)(4,6)(5,8)(17,18,23,26)(19,27,31,45)(20,33,35,25)(21,38,32,24)(22,40,29,28)(30,43,34,46)(36,42,39,41)(37,44,47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,29,21,23,31,22,32)(18,24,40,45,26,38,28,27)(20,34,47,36,35,30,37,39)(25,42,48,46,33,41,44,43);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,39)(22,37)(23,35)(24,41)(26,33)(27,46)(28,44)(29,47)(31,34)(32,36)(38,42)(40,48)(43,45);(2,6)(5,7)(10,14)(13,16)(18,26)(19,31)(20,35)(21,32)(28,40)(37,47)(41,42)(43,46);(2,6)(5,7)(17,21,29,31,23,32,22,19)(18,27,40,24,26,45,28,38)(20,36,47,30,35,39,37,34)(25,43,48,42,33,46,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,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, 7507705);