#7361244

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1641 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)(19,24,22,37)(20,30,33,38)(21,26,35,43)(25,31,41,28)(29,39,36,48)(32,46,34,47)(40,45,42,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,30,41,38)(26,39,43,48)(32,45,34,44)(40,46,42,47);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,34)(22,31)(23,33)(24,38)(26,42)(27,41)(29,45)(30,37)(32,35)(36,44)(39,46)(40,43)(47,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,29)(20,32)(22,36)(23,35)(24,39)(25,40)(27,43)(28,44)(30,46)(31,45)(33,34)(37,48)(38,47)(41,42);(2,6)(5,7)(10,14)(13,15)(20,33)(21,35)(25,41)(26,43)(28,31)(29,36)(30,38)(39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,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, 7268440);