#7045924

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1209 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6953120);