#8269725

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

4220 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,24)(19,27,29,23)(20,30,31,25)(21,26,32,28)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(2,6)(7,9)(8,13)(14,15)(17,19,22,29)(18,23,24,27)(20,26,31,28)(21,25,32,30)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(4,12)(8,13)(10,16)(14,15)(18,24)(21,32)(23,27)(25,30)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,27)(26,31)(30,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,31)(18,25,24,30)(19,28,29,26)(21,27,32,23)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21)(18,26)(19,25)(20,27)(22,32)(23,31)(24,28)(29,30)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=ghi,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, 8176921);