#7608225

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ghi,d^2=g,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, 7515421);