#7230230

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1123 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,25,30)(20,27,29,23)(26,32,28,31)(33,34,37,36)(35,41,43,48)(38,45,46,40)(39,47,42,44);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19)(18,22)(20,28)(21,25)(23,32)(24,30)(26,29)(27,31)(34,36)(39,42)(40,45)(41,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(33,35,38,44,37,43,46,47)(34,39,40,41,36,42,45,48);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,20,21,29)(18,23,24,27)(19,26,25,28)(22,31,30,32)(33,36,37,34)(35,42,43,39)(38,45,46,40)(41,44,48,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,25)(22,30)(26,28)(31,32)(34,36)(39,42)(40,45)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fgi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7137426);