#8417105

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	#52418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8324301);