#8492914

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	#40692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=i,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, 8400110);