#8457049

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	#55699	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8364245);