#8436220

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	#36987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,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, 8343416);