#7614718

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,34)(21,37)(22,27)(23,28)(24,32)(25,31)(26,40)(30,42)(33,45)(35,44)(36,48)(38,46)(39,47)(41,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,35,31,30)(21,38,40,36)(23,41,39,33)(25,44,34,42)(26,46,37,48)(28,45,47,43);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,30)(21,33)(22,31)(23,36)(24,42)(26,45)(27,34)(28,46)(29,44)(32,35)(37,43)(38,39)(40,41)(47,48);(10,14)(13,15)(17,21,39,30,22,40,23,35)(18,26,47,44,27,37,28,42)(19,31,41,36,32,20,33,38)(24,25,45,48,29,34,43,46);(2,5)(6,8)(10,14)(13,15)(18,27)(20,31)(21,40)(24,29)(28,47)(30,35)(36,38)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fg,d^2=hi,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, 7521914);