#8679893

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,15)(14,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34)(35,41)(36,46)(37,43)(38,39)(40,44)(42,45)(47,48);(1,3)(2,9)(4,6)(5,14)(7,11)(8,15)(10,16)(12,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,40,38)(34,41,44,39)(36,47,45,37)(42,43,46,48);(1,4,8,5)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,36)(34,42)(35,37)(38,47)(39,43)(40,45)(41,48)(44,46);(1,5,8,4)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,37,40,47)(34,43,44,48)(35,45,38,36)(39,46,41,42);(1,6,8,14)(2,11,12,16)(3,4,15,5)(7,13,10,9)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,38,40,35)(34,41,44,39)(36,47,45,37)(42,48,46,43);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,9,14,13)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,39,40,41)(34,38,44,35)(36,43,45,48)(37,42,47,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587089);