#8115397

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022593);