#8492778

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=g,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, 8399974);