#7052213

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(19,24,22,37)(20,30,32,46)(21,26,35,42)(25,28,40,33)(29,38,36,48)(31,45,34,47)(39,43,41,44);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,23,22)(18,24,27,37)(20,28,32,33)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,22,33,23,32,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,21,23,35)(18,26,27,42)(19,29,22,36)(20,31,32,34)(24,38,37,48)(25,39,40,41)(28,43,33,44)(30,45,46,47);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(20,32)(21,35)(25,40)(26,42)(28,33)(29,36)(30,46)(38,48)

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=dhi,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, 6959409);