#8898864

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806060);