#7171404

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	#45243	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(10,13)(11,15)(17,18,21,27,22,26,33,24)(19,23,30,34,28,36,42,37)(20,25,32,39,31,40,44,38)(29,35,43,45,41,47,48,46);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19,22,28)(18,23,26,36)(20,29,31,41)(21,30,33,42)(24,37,27,34)(25,35,40,47)(32,43,44,48)(38,46,39,45);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(3,8)(6,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(28,41)(30,43)(33,44)(34,45)(36,47)(37,46)(42,48);(9,11,12,15)(10,14,13,16)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,47)(45,46)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7078600);