#8718359

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	#57004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(3,8)(4,10)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34,39,41)(35,43,37,40)(36,38,44,42)(45,46,47,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,10)(12,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,39,37)(34,40,41,43)(36,47,44,45)(38,46,42,48);(2,9)(3,4)(11,14)(12,13)(20,29)(23,28)(26,32)(30,31)(34,41)(36,44)(40,43)(45,47);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32)(33,36)(34,42)(35,45)(37,47)(38,41)(39,44)(40,46)(43,48);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,37,39,35)(34,43,41,40)(36,47,44,45)(38,46,42,48);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,38,39,42)(34,44,41,36)(35,46,37,48)(40,45,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=i,e^2=h,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, 8625555);