#8247674

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	#54850	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^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, 8154870);