#7149676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056872);