#7614797

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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(19,27)(20,31)(23,28)(24,38)(30,41)(32,46)(34,39)(37,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,25)(19,28,27,23)(20,32,31,46)(22,35,33,26)(24,39,38,34)(29,44,40,47)(30,37,41,43)(36,45,42,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,27)(18,23,25,28)(20,29,31,40)(22,30,33,41)(24,36,38,42)(26,37,35,43)(32,47,46,44)(34,45,39,48);(10,14)(13,15)(17,20,33,39,21,31,22,34)(18,24,35,46,25,38,26,32)(19,29,41,48,27,40,30,45)(23,36,43,44,28,42,37,47);(2,5)(6,8)(10,14)(13,15)(19,27)(23,28)(29,40)(30,41)(36,42)(37,43)(44,47)(45,48)

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

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, 7521993);