#8271229

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8178425);