#7361767

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	#45243	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 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)(17,18,24,29)(19,25,33,42)(20,27,21,35)(22,28,23,38)(26,37,43,34)(30,44,36,40)(31,41,32,47)(39,46,48,45);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,33)(26,39)(27,44)(28,41)(29,42)(34,45)(35,40)(37,46)(38,47)(43,48);(9,11,12,16)(10,13,14,15)(17,20,24,21)(18,26,29,43)(19,30,33,36)(22,34,23,37)(25,39,42,48)(27,28,35,38)(31,45,32,46)(40,47,44,41);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,24,20)(18,27,29,35)(19,30,33,36)(22,34,23,37)(25,40,42,44)(26,28,43,38)(31,46,32,45)(39,47,48,41);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,33,32)(20,34,21,37)(25,41,42,47)(26,35,43,27)(30,45,36,46)(39,40,48,44)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fi,e^2=i,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, 7268963);