#8421823

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4496 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329019);