#8493189

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	#53742	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=ghi,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, 8400385);