#10340249

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	#54056	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(17,18,23,21)(19,24,26,20)(22,29,31,25)(27,28,30,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(2,7)(3,11)(5,14)(12,15)(17,19)(18,24)(20,21)(22,30)(23,26)(25,28)(27,31)(29,32);(2,7)(4,9)(8,10)(12,15)(18,21)(19,26)(25,29)(27,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,21,23,18)(19,20,26,24)(22,29,31,25)(27,28,30,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=h,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, 10247445);