#8720371

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,22,23)(19,20,26,25)(21,24,31,32)(27,28,29,30)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,48,45);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18,22,23)(19,25,26,20)(21,24,31,32)(27,30,29,28)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(4,13)(8,12)(11,14)(15,16)(18,23)(19,26)(24,32)(27,29)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19)(18,20)(21,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,36,39,47)(34,41,44,46)(35,45,37,43)(38,42,48,40);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,37,39,35)(34,42,44,40)(36,43,47,45)(38,41,48,46);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,21,22,31)(18,24,23,32)(19,27,26,29)(20,28,25,30)(33,38,39,48)(34,43,44,45)(35,41,37,46)(36,40,47,42)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=hi,e^2=g,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, 8627567);