#8115336

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,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, 8022532);