#10324605

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	#40917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4264 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30)(33,34,38,40)(35,43,45,39)(36,46,47,41)(37,42,48,44);(2,3)(5,13)(7,9)(12,16)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27)(33,35,38,45)(34,39,40,43)(36,42,47,44)(37,41,48,46);(4,11)(7,9)(12,16)(14,15)(18,23)(19,25)(24,29)(26,30)(34,40)(37,48)(39,43)(41,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)(33,36,38,47)(34,41,40,46)(35,44,45,42)(37,43,48,39);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,37)(34,42)(35,41)(36,43)(38,48)(39,47)(40,44)(45,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)(33,35)(34,39)(36,44)(37,41)(38,45)(40,43)(42,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^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, 10231801);