#8425434

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	#40692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332630);