#8253890

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gh,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161086);