#8105905

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013101);