#6719795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6626991);