#8269713

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	#59729	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=ghi,e^2=ghi,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, 8176909);