#8463641

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=g,e^2=ghi,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, 8370837);