#7312346

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1682 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)(3,6)(4,5)(7,8)(17,18,21,26,22,25,35,41)(19,27,30,36,28,42,34,23)(20,24,33,40,31,38,45,48)(29,43,44,39,32,46,47,37);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(19,28)(20,31)(23,36)(24,38)(27,42)(30,34)(33,45)(40,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,25,36)(20,32,31,29)(21,34,35,30)(24,39,38,37)(26,42,41,27)(33,44,45,47)(40,43,48,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,31)(18,24,25,38)(19,29,28,32)(21,33,35,45)(23,37,36,39)(26,40,41,48)(27,43,42,46)(30,44,34,47);(2,5)(6,8)(10,14)(13,15)(18,25)(20,31)(23,36)(26,41)(27,42)(29,32)(33,45)(44,47)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,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, 7219542);