#2608389

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2204 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2515585);