#8456946

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

3978 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=h,e^2=ghi,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, 8364142);