#10407828

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315024);