#8463772

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(18,20)(23,26)(24,28)(30,32)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(19,25)(23,26)(27,31)(30,32)(34,40)(35,45)(36,42)(43,47);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=hi,f^2=ghi,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, 8370968);