#8464380

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	#45600	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34)(35,40)(36,41)(37,47)(38,43)(39,44)(42,46)(45,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,48)(46,47);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,35,39,48)(34,40,44,45)(36,46,43,37)(38,47,41,42);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,36)(34,41)(35,46)(37,48)(38,44)(39,43)(40,42)(45,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,45,41,40);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,45,46,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=hi,f^2=gi,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, 8371576);