#8271072

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,15)(14,16)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(3,10)(4,13)(14,15)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(34,35)(39,41)(40,42)(46,47);(2,7)(5,8)(9,12)(14,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,6,13)(2,9,7,12)(3,11,10,16)(5,14,8,15)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,8)(2,3,7,10)(4,14,13,15)(9,11,12,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=ghi,e^2=hi,f^2=gi,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, 8178268);