#8457050

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	#54272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 8364246);