#10529823

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	#50787	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10437019);