#8456002

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=g,e^2=gi,f^2=g,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, 8363198);