#8455663

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8362859);