#7045432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6952628);