#8270592

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

3881 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gi,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, 8177788);