#8425630

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,e^2=gh,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, 8332826);