#7607026

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

1344 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)(9,10,13,12)(11,14,16,15)(17,18)(19,26)(20,25)(21,35)(22,38)(23,28)(24,31)(27,37)(29,43)(30,44)(32,40)(33,39)(34,47)(36,48)(41,45)(42,46);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,31,23,30,37,21)(18,24,27,26,28,35,38,44)(20,29,34,33,32,45,46,36)(25,39,42,43,40,48,47,41);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,29)(21,36)(22,34)(23,32)(24,39)(26,43)(27,42)(28,40)(30,45)(31,33)(35,48)(37,46)(38,47)(41,44);(2,6)(5,7)(11,16)(14,15)(19,30)(20,32)(21,31)(24,35)(25,40)(26,44)(34,46)(42,47);(2,6)(5,7)(17,21,37,30,23,31,22,19)(18,26,38,24,28,44,27,35)(20,33,46,29,32,36,34,45)(25,41,47,48,40,43,42,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=hi,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,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, 7514222);