#7170910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7078106);