#7208453

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	#53064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,38)(20,30,33,40)(21,34,37,26)(25,28,42,31)(29,43,35,39)(32,48,36,47)(41,46,44,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,38)(20,31,33,28)(21,35,37,29)(25,40,42,30)(26,43,34,39)(32,45,36,46)(41,47,44,48);(2,6)(5,7)(11,15)(13,16)(17,20,22,31,23,33,19,28)(18,25,24,30,27,42,38,40)(21,36,35,45,37,32,29,46)(26,44,39,48,34,41,43,47);(3,8)(5,7)(10,14)(11,15)(17,21,23,37)(18,26,27,34)(19,29,22,35)(20,32,33,36)(24,39,38,43)(25,41,42,44)(28,45,31,46)(30,47,40,48);(2,6)(5,7)(11,15)(13,16)(20,33)(21,37)(25,42)(26,34)(28,31)(29,35)(30,40)(39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=i,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=ehi,d^c=di,e^c=ei,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, 7115649);