#8630846

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	#36735	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,8,13,15)(4,10,11,5)(6,12,16,14)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34,39,42)(35,43,41,37)(36,46,38,40)(44,48,45,47);(1,3)(2,8)(4,14)(5,16)(6,10)(7,13)(9,15)(11,12)(18,23)(19,26)(24,30)(27,32)(33,35,39,41)(34,37,42,43)(36,44,38,45)(40,48,46,47);(2,9)(3,13)(5,10)(12,14)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,36,39,38)(34,40,42,46)(35,44,41,45)(37,48,43,47);(1,4,7,11)(2,10,9,5)(3,12,13,14)(6,8,16,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,47)(45,48);(1,5,7,10)(2,11,9,4)(3,6,13,16)(8,14,15,12)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37,39,43)(34,41,42,35)(36,47,38,48)(40,44,46,45);(1,6)(2,12)(3,10)(4,15)(5,13)(7,16)(8,11)(9,14)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,38,39,36)(34,40,42,46)(35,45,41,44)(37,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538042);