#8993527

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	#53426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900723);