#7172329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1271 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)(10,13)(11,15)(17,18,23,25,21,27,22,28)(19,29,36,40,31,24,32,38)(20,26,37,41,34,43,35,44)(30,42,46,48,33,39,45,47);(1,2,5,6)(3,7,4,8)(10,13)(11,15)(17,19,21,31)(18,24,27,29)(20,33,34,30)(22,32,23,36)(25,38,28,40)(26,42,43,39)(35,46,37,45)(41,48,44,47);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(18,25)(22,23)(24,38)(26,41)(27,28)(29,40)(32,36)(35,37)(39,47)(42,48)(43,44)(45,46);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,37)(24,39)(25,41)(27,43)(28,44)(29,42)(31,33)(32,45)(36,46)(38,47)(40,48);(9,11,12,15)(10,14,13,16)(18,27)(20,34)(24,29)(25,28)(30,33)(35,37)(38,40)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=fi,d^2=fi,e^2=h,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, 7079525);