#8458679

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	#59401	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(3,7)(4,14)(5,9)(6,10)(8,11)(12,16)(13,15)(17,18)(19,29)(20,26)(21,23)(22,32)(24,28)(25,31)(27,30)(36,44)(40,47)(43,48)(45,46);(1,3,8,6)(2,7,11,10)(4,13,5,12)(9,16,14,15)(17,19,24,31)(18,25,28,29)(20,30,21,22)(23,27,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,8,5)(2,9,11,14)(3,12,6,13)(7,15,10,16)(17,20,24,21)(18,26,28,23)(19,30,31,22)(25,32,29,27)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,5,8,4)(2,9,11,14)(3,13,6,12)(7,15,10,16)(17,21,24,20)(18,26,28,23)(19,22,31,30)(25,32,29,27)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,6,8,3)(2,10,11,7)(4,13,5,12)(9,16,14,15)(17,22)(18,27)(19,20)(21,31)(23,29)(24,30)(25,26)(28,32)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,7,8,10)(2,3,11,6)(4,15,5,16)(9,12,14,13)(17,23,24,26)(18,21,28,20)(19,27,31,32)(22,29,30,25)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=i,f^2=gi,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, 8365875);