#8425342

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,f^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, 8332538);