#6937400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#30722	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844596);