#8456978

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	#42447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,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, 8364174);