#7355356

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#53152	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fgi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,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, 7262552);