#7186200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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)(9,10,13,16)(11,17,18,14)(12,20,22,15)(19,23,21,24)(25,26)(27,35)(28,39)(29,34)(30,36)(31,32)(33,42)(37,49)(38,50)(40,55)(41,46)(43,47)(44,56)(45,52)(48,54)(51,53);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,17)(12,21,22,19)(15,24,20,23)(25,27,30,31)(26,32,36,35)(28,37,41,42)(29,43,45,38)(33,46,49,39)(34,50,52,47)(40,54,44,53)(48,55,51,56);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,28,31,42,30,41,27,37)(26,33,35,39,36,49,32,46)(29,44,43,53,45,40,38,54)(34,51,50,56,52,48,47,55);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,29)(26,34)(27,38)(28,40)(30,45)(31,43)(32,47)(33,48)(35,50)(36,52)(37,53)(39,55)(41,44)(42,54)(46,56)(49,51);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(25,27,30,31)(26,35,36,32)(28,37,41,42)(29,43,45,38)(33,39,49,46)(34,47,52,50)(40,54,44,53)(48,56,51,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fgi,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=eh,d^c=dhi,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, 7093396);