#8097054

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	#45560	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004250);