#8172612

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8079808);