#7616074

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

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

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523270);