#3547096

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fgi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454292);