#8094666

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001862);