#8508969

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416165);