#7301385

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

1661 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)(17,18,23,27)(19,28,29,44)(20,25,35,43)(21,26,22,37)(24,31,38,32)(30,45,33,48)(34,41,36,42)(39,47,40,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(19,29)(24,38)(28,44)(30,33)(31,32)(39,40)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,38)(20,33,35,30)(21,32,22,31)(25,40,43,39)(26,44,37,28)(34,46,36,47)(41,48,42,45);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,39)(26,42)(27,43)(28,45)(29,33)(31,47)(32,46)(37,41)(38,40)(44,48);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(24,28,38,44)(25,41,43,42)(30,46,33,47)(39,48,40,45)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^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,e^c=ei,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, 7208581);