#8724768

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=h,f^2=gi,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, 8631964);