#8901339

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

3927 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8808535);