#8445656

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

4064 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,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, 8352852);