#8174319

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	#53102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081515);