#8456823

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4646 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26,31,34)(27,35,38,30)(28,29,33,32)(36,39,40,37);(3,4)(6,8)(10,15)(11,18)(16,22)(19,24)(25,27,31,38)(26,30,34,35)(28,36,33,40)(29,37,32,39);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28,31,33)(26,32,34,29)(27,36,38,40)(30,39,35,37);(1,4,5,3)(2,6,7,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,26)(27,35)(28,32)(29,33)(30,38)(31,34)(36,37)(39,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,29,31,32)(26,33,34,28)(27,37,38,39)(30,40,35,36);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,30,31,35)(26,27,34,38)(28,39,33,37)(29,40,32,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gh,e^2=hi,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, 8364019);