#10324567

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	#54820	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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=g,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, 10231763);