#8463860

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

4048 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)(3,10,6,9)(4,7,5,15)(12,16,13,14)(19,20)(23,24)(27,28)(30,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,8,6)(2,9,11,10)(4,12,5,13)(7,14,15,16)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(34,39)(35,42)(41,44)(43,48);(1,4,8,5)(2,7,11,15)(3,12,6,13)(9,14,10,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5,8,4)(2,7,11,15)(3,13,6,12)(9,14,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,8,3)(2,10,11,9)(4,13,5,12)(7,16,15,14)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,7)(2,5)(3,14)(4,11)(6,16)(8,15)(9,13)(10,12)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 8371056);