#8517612

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	#40669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,2,6,7)(3,8,13,16)(4,9,14,12)(5,10,15,11)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(34,39)(35,41)(40,46)(42,47);(2,7)(4,14)(8,16)(10,11)(17,19,21,27)(18,23,25,22)(20,26,29,32)(24,31,28,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3)(2,8)(4,11)(5,9)(6,13)(7,16)(10,14)(12,15)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4)(2,9)(3,11)(5,8)(6,14)(7,12)(10,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48);(1,5)(2,10)(3,12)(4,8)(6,15)(7,11)(9,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424808);