#7051893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6959089);