#8345917

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3938 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253113);