#7201612

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,32,41)(20,33,21,26)(22,36,37,27)(25,35,42,34)(29,43,30,39)(31,46,47,40)(38,45,48,44);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,28,41)(20,29,21,30)(22,31,37,47)(25,38,42,48)(26,39,33,43)(27,40,36,46)(34,44,35,45);(11,15)(13,16)(17,20,23,21)(18,25,28,42)(19,29,32,30)(22,35,37,34)(24,38,41,48)(26,36,33,27)(31,45,47,44)(39,46,43,40);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,21,23,20)(18,26,28,33)(19,30,32,29)(22,35,37,34)(24,39,41,43)(25,36,42,27)(31,45,47,44)(38,46,48,40);(2,5)(6,8)(11,15)(13,16)(20,21)(25,42)(26,33)(29,30)(34,35)(38,48)(39,43)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7108808);