#8445006

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	#39996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=ghi,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, 8352202);