#8436733

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	#56409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=h,f^2=g,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, 8343929);