#7589359

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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)(17,18,21,25)(19,27,30,23)(20,24,33,39)(22,26,35,41)(28,38,32,36)(29,37,31,42)(34,40,45,48)(43,46,44,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,32,33,28)(22,29,35,31)(24,38,39,36)(26,42,41,37)(34,44,45,43)(40,46,48,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,33)(18,24,25,39)(19,28,30,32)(22,34,35,45)(23,36,27,38)(26,40,41,48)(29,43,31,44)(37,46,42,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,25)(19,29)(22,35)(23,37)(24,39)(27,42)(28,43)(30,31)(32,44)(34,45)(36,46)(38,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,25)(20,33)(23,27)(26,41)(28,32)(34,45)(37,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,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, 7496555);