#8458823

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	#42602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(20,28)(24,31)(27,32)(29,30)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,48,46);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,40,47)(34,41,38,45)(36,46,37,44)(39,43,48,42);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,40,37)(34,42,38,43)(35,46,47,44)(39,45,48,41);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37,40,36)(34,43,38,42)(35,46,47,44)(39,45,48,41);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,46,48,44);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,39,40,48)(34,44,38,46)(35,43,47,42)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=gi,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, 8366019);