#8467008

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(17,18,23,22)(19,24,29,27)(20,25,30,32)(21,31,28,26)(33,34,40,38)(35,44,47,41)(36,48,42,37)(39,43,46,45);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18,23,22)(19,27,29,24)(20,25,30,32)(21,26,28,31)(33,35)(34,41)(36,45)(37,39)(38,44)(40,47)(42,43)(46,48);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19,23,29)(18,24,22,27)(20,28,30,21)(25,26,32,31)(33,36)(34,37)(35,45)(38,48)(39,41)(40,42)(43,47)(44,46);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,20)(18,25)(19,28)(21,29)(22,32)(23,30)(24,26)(27,31)(33,37,40,48)(34,42,38,36)(35,46,47,39)(41,45,44,43);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,21,23,28)(18,26,22,31)(19,20,29,30)(24,32,27,25)(33,38,40,34)(35,44,47,41)(36,48,42,37)(39,45,46,43);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,22,23,18)(19,27,29,24)(20,25,30,32)(21,31,28,26)(33,39,40,46)(34,43,38,45)(35,37,47,48)(36,41,42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=g,e^2=hi,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, 8374204);