#8471129

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

4180 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(17,18,23,20)(19,24,30,27)(21,31,29,25)(22,26,28,32)(33,34,39,43)(35,44,45,40)(36,41,46,48)(37,38,42,47);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(35,45)(36,46)(40,44)(41,48);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,18,23,20)(19,27,30,24)(21,25,29,31)(22,26,28,32)(33,35)(34,40)(36,42)(37,46)(38,41)(39,45)(43,44)(47,48);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,20,23,18)(19,27,30,24)(21,31,29,25)(22,26,28,32)(33,36,39,46)(34,41,43,48)(35,42,45,37)(38,44,47,40);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32)(33,37,39,42)(34,38,43,47)(35,46,45,36)(40,41,44,48);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)(33,38)(34,42)(35,41)(36,40)(37,43)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8378325);