#10324624

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	#46633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,7)(3,11)(5,14)(12,15)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27)(35,43)(37,47)(39,48)(41,44);(2,7)(4,9)(8,10)(12,15)(18,23)(19,25)(24,29)(26,30)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 10231820);