#7301320

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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1659 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,36,25)(23,35,37,30)(29,43,31,45)(33,48,47,41)(38,46,39,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,42)(29,31)(30,35)(38,39)(43,45)(44,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,37)(20,31,32,29)(21,35,36,30)(24,39,40,38)(25,42,34,27)(33,44,47,46)(41,43,48,45);(3,8)(5,7)(10,14)(11,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,38)(25,41)(26,40)(27,43)(28,31)(30,44)(34,48)(35,46)(36,47)(37,39)(42,45);(2,6)(5,7)(10,14)(13,15)(20,32)(24,40)(29,31)(33,47)(38,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7208516);