#8202178

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

3984 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,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, 8109374);