#7211429

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,36)(28,39)(29,34)(30,35)(31,32)(33,42)(37,48)(38,49)(40,52)(41,45)(43,46)(44,50)(47,53)(51,56)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,43,44,38)(33,45,48,39)(34,49,50,46)(40,53,54,51)(47,52,56,55);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,37,30,41,31,42)(26,33,32,45,35,48,36,39)(29,40,38,51,44,54,43,53)(34,47,46,55,50,56,49,52);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29)(26,34)(27,38)(28,40)(30,44)(31,43)(32,46)(33,47)(35,50)(36,49)(37,51)(39,52)(41,54)(42,53)(45,55)(48,56);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,35)(29,44)(32,36)(33,48)(38,43)(39,45)(40,54)(51,53)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7118625);