#8664423

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	#53177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8571619);