#7600655

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(17,18,23,28)(19,29,32,26)(20,25,33,41)(21,24,37,40)(22,38,30,27)(31,42,46,44)(34,47,36,39)(35,48,45,43);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,30,37,23,32,22,21)(18,24,38,26,28,40,27,29)(20,31,45,34,33,46,35,36)(25,39,48,44,41,47,43,42);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,31)(21,36)(22,35)(23,33)(24,39)(26,44)(27,43)(28,41)(29,42)(30,45)(32,46)(34,37)(38,48)(40,47);(2,6)(3,8)(10,14)(13,16)(19,32)(20,33)(21,37)(24,40)(25,41)(26,29)(35,45)(43,48);(2,6)(5,7)(17,21,22,32,23,37,30,19)(18,26,27,24,28,29,38,40)(20,34,35,31,33,36,45,46)(25,42,43,47,41,44,48,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,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,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, 7507851);