#7529270

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1460 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7436466);