#8095069

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	#52963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002265);