#7338492

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#31737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=f,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^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, 7245688);