#8492816

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

4229 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)(3,9,4,8)(5,14,15,10)(6,11,16,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,44)(36,42)(37,41)(38,40)(39,43)(45,47)(46,48);(1,3,7,4)(2,8,12,9)(5,11,15,13)(6,10,16,14)(19,26)(20,28)(22,23)(24,31)(33,35,39,38)(34,40,43,44)(36,45,37,46)(41,47,42,48);(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,4,7,3)(2,9,12,8)(5,11,15,13)(6,10,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37,39,36)(34,42,43,41)(35,45,38,46)(40,47,44,48);(1,5)(2,10)(3,13)(4,11)(6,8)(7,15)(9,16)(12,14)(17,20,21,28)(18,24,25,31)(19,27,26,32)(22,29,23,30)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,48)(46,47);(1,6,7,16)(2,11,12,13)(3,10,4,14)(5,8,15,9)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,31)(30,32)(33,38,39,35)(34,40,43,44)(36,46,37,45)(41,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=g,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, 8400012);