#10439966

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	#54195	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18,23,26)(19,24,21,29)(20,22,25,32)(27,28,30,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(19,21)(24,29)(27,30)(28,31)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,26,29)(20,27,25,30)(22,28,32,31)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20)(18,22)(19,27)(21,30)(23,25)(24,28)(26,32)(29,31)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,23,19)(18,24,26,29)(20,27,25,30)(22,31,32,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22)(18,25)(19,28)(20,26)(21,31)(23,32)(24,30)(27,29)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)

Polycyclic presentation:

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

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, 10347162);