#8436816

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	#41805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,31,30)(33,34,39,43)(35,44,45,40)(36,41,47,46)(37,38,42,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32)(35,45)(36,47)(40,44)(41,46);(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,35,39,45)(34,40,43,44)(36,42,47,37)(38,46,48,41);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32)(33,36)(34,41)(35,42)(37,45)(38,40)(39,47)(43,46)(44,48);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,27,31,23)(33,37,39,42)(34,38,43,48)(35,36,45,47)(40,46,44,41);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,27,29,23)(33,38)(34,42)(35,46)(36,44)(37,43)(39,48)(40,47)(41,45)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,e^2=h,f^2=g,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, 8344012);