#7302944

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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1809 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)(17,18,23,27)(19,24,22,36)(20,30,32,38)(21,33,35,26)(25,37,41,28)(29,42,34,39)(31,45,47,44)(40,43,48,46);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,34,35,29)(25,38,41,30)(26,42,33,39)(31,46,47,43)(40,44,48,45);(9,11)(10,13)(12,16)(14,15)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,35,47)(24,38,36,30)(26,40,33,48)(29,43,34,46)(39,45,42,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21)(18,26)(19,29)(20,31)(22,34)(23,35)(24,39)(25,40)(27,33)(28,43)(30,44)(32,47)(36,42)(37,46)(38,45)(41,48);(2,5)(6,8)(10,14)(13,15)(21,35)(26,33)(29,34)(31,47)(39,42)(40,48)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,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, 7210140);