#8492940

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	#56717	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=gi,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, 8400136);