#8092253

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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	#52427	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,27)(26,31,33,35)(29,34,37,36)(32,39,40,38);(3,7)(5,8)(10,14)(11,17)(16,21)(18,23)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39);(2,6)(5,8)(10,14)(15,20)(16,21)(22,24)(26,33)(31,35)(32,40)(38,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

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, 7999449);