#7330763

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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

1433 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7237959);