#7160700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7067896);