#8455957

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=g,e^2=i,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, 8363153);