#8463866

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	#54002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,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, 8371062);