#8720366

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,22,23)(19,20,26,25)(21,24,31,32)(27,28,29,30)(33,34,39,43)(35,36,46,40)(37,47,45,41)(38,42,44,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18,22,23)(19,25,26,20)(21,24,31,32)(27,30,29,28)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,48)(42,47);(4,13)(7,12)(10,16)(14,15)(18,23)(19,26)(24,32)(27,29)(33,36,39,40)(34,35,43,46)(37,42,45,48)(38,41,44,47);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,19)(18,20)(21,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,36)(34,35)(37,42)(38,47)(39,40)(41,44)(43,46)(45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,37)(34,41)(35,44)(36,42)(38,46)(39,45)(40,48)(43,47);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,21,22,31)(18,24,23,32)(19,27,26,29)(20,28,25,30)(33,38,39,44)(34,42,43,48)(35,45,46,37)(36,41,40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627562);