#8720384

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	#41300	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(3,11)(4,13)(5,8)(6,12)(7,9)(10,15)(14,16)(17,18,22,23)(19,20,26,25)(21,24,31,32)(27,28,29,30)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,10,16,13)(17,18,22,23)(19,25,26,20)(21,24,31,32)(27,30,29,28)(34,35)(39,41)(40,42)(46,47);(2,9)(4,15)(8,11)(12,14)(18,23)(19,26)(24,32)(27,29)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4)(2,10)(3,12)(5,14)(6,8)(7,15)(9,13)(11,16)(17,19)(18,20)(21,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,10,16,13)(17,20)(18,19)(21,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6,7,16)(2,12,9,14)(3,10,5,13)(4,11,15,8)(17,21,22,31)(18,24,23,32)(19,27,26,29)(20,28,25,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=ghi,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, 8627580);