#7184005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=f,e^2=h,f^2=i,g^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,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7091201);