#7173226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1334 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,32,38)(27,39,44,50)(28,34,29,35)(30,37,31,36)(33,42,51,43)(40,55,46,49)(41,53,56,48)(45,54,47,52);(1,3,4,7)(2,5,6,8)(11,17)(12,20)(14,21)(15,19)(25,27,31,43,32,44,30,42)(26,33,37,39,38,51,36,50)(28,40,45,56,29,46,47,41)(34,48,52,55,35,53,54,49);(2,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,32,29)(26,34,38,35)(27,40,44,46)(30,47,31,45)(33,48,51,53)(36,54,37,52)(39,55,50,49)(41,43,56,42);(11,17)(14,21)(18,23)(22,24)(25,29,32,28)(26,35,38,34)(27,41,44,56)(30,47,31,45)(33,49,51,55)(36,54,37,52)(39,53,50,48)(40,43,46,42);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,32,31)(26,36,38,37)(27,42,44,43)(28,45,29,47)(33,50,51,39)(34,52,35,54)(40,56,46,41)(48,55,53,49)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080422);