#8445685

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

4047 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(34,36)(39,41)(40,44)(46,47);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=i,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, 8352881);