#8504805

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412001);