#8463748

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,20)(23,26)(24,28)(30,32)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(19,25)(23,26)(27,31)(30,32)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=hi,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, 8370944);