#8471150

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

4153 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,23,20)(19,24,30,27)(21,31,29,25)(22,26,28,32)(33,34,40,44)(35,41,38,39)(36,46,45,42)(37,43,47,48);(1,3)(2,8)(4,6)(5,13)(7,14)(9,10)(11,12)(15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,35,40,38)(34,41,44,39)(36,47,45,37)(42,43,46,48);(1,4,7,5)(2,9,11,15)(3,6,14,13)(8,10,12,16)(17,18,23,20)(19,27,30,24)(21,25,29,31)(22,26,28,32)(33,36)(34,42)(35,37)(38,47)(39,43)(40,45)(41,48)(44,46);(1,5,7,4)(2,9,11,15)(3,13,14,6)(8,10,12,16)(17,20,23,18)(19,27,30,24)(21,31,29,25)(22,26,28,32)(33,37,40,47)(34,43,44,48)(35,45,38,36)(39,46,41,42);(1,6,7,13)(2,10,11,16)(3,4,14,5)(8,9,12,15)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32)(33,38,40,35)(34,39,44,41)(36,47,45,37)(42,43,46,48);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)(33,39)(34,35)(36,43)(37,42)(38,44)(40,41)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378346);