#10324560

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	#48375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=g,e^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, 10231756);