#7548619

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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,35)(28,42)(29,37)(30,36)(31,38)(32,39)(33,40)(34,41)(43,47)(44,46)(45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,31)(21,28)(22,29)(24,38)(25,37)(26,42)(27,32)(30,34)(33,44)(35,39)(36,41)(40,46)(43,45)(47,48);(10,14)(11,16)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,39)(26,40)(28,43)(29,31)(30,44)(34,45)(36,46)(37,38)(41,48)(42,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(19,28)(21,34)(23,36)(24,39)(27,43)(29,30)(31,44)(33,45)(35,46)(37,42)(38,47);(2,5)(6,8)(10,14)(13,15)(19,29)(20,32)(23,37)(24,39)(28,30)(33,45)(36,42)(40,48)

Polycyclic presentation:

2^9|d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,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, 7455815);