#7187141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1341 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fg,d^2=fi,e^2=hi,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=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094337);