#8110606

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

4616 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239255, #72239258
maximal normal subgroup of	#72239255, #72239258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017802);