#8093252

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	#46287	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000448);