#7045032

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,33)(29,34)(30,31)(32,40)(35,45)(37,53)(38,46)(39,51)(41,47)(42,56)(43,50)(44,52)(48,54)(49,55);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(28,41)(32,46)(34,35)(36,51)(37,42)(43,44)(54,55);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,40,30,39,45,38)(26,32,35,36,31,46,34,51)(28,42,44,55,41,37,43,54)(33,49,50,56,47,48,52,53);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,41)(26,33,31,47)(27,37,39,42)(29,44,45,43)(32,48,46,49)(34,52,35,50)(36,53,51,56)(38,55,40,54);(10,16)(11,18)(15,24)(17,19)(25,29,30,45)(26,34,31,35)(27,38,39,40)(28,43,41,44)(32,36,46,51)(33,50,47,52)(37,54,42,55)(48,56,49,53)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgi,d^2=f,e^2=i,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, 6952228);