#7303479

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(17,18,22,28)(19,30,33,24)(20,35,37,26)(21,29,23,27)(25,38,43,36)(31,40,47,45)(32,41,34,42)(39,46,44,48);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19,22,33)(18,24,28,30)(20,31,37,47)(21,32,23,34)(25,39,43,44)(26,40,35,45)(27,41,29,42)(36,46,38,48);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,29)(27,35)(28,43)(30,44)(32,46)(33,47)(34,48)(40,42)(41,45);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,40)(25,29)(27,43)(28,35)(30,45)(32,48)(33,47)(34,46)(39,42)(41,44);(2,6)(5,7)(10,11)(14,15)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,36,37,38)(24,41,30,42)(25,35,43,26)(31,46,47,48)(39,45,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7210675);