#8271292

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

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

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, 8178488);