#7600926

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,29,21,37)(20,33,35,25)(22,39,30,28)(24,38,27,32)(31,45,34,43)(36,42,44,47)(40,46,41,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,30,38,23,21,22,32)(18,24,39,37,26,27,28,29)(20,34,44,46,35,31,36,48)(25,41,47,43,33,40,42,45);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20)(18,25)(19,31)(21,34)(22,36)(23,35)(24,40)(26,33)(27,41)(28,42)(29,43)(30,44)(32,46)(37,45)(38,48)(39,47);(2,5)(6,7)(10,14)(11,16)(18,26)(20,35)(24,27)(28,39)(29,37)(31,34)(36,44)(46,48);(2,5)(6,7)(10,14)(13,15)(17,21,22,38,23,19,30,32)(18,27,28,37,26,24,39,29)(20,34,36,48,35,31,44,46)(25,41,42,45,33,40,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,e^2=fgi,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508122);