#8094601

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	#52973	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001797);