#4117646

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303941, #72304088, #72304090
maximal normal subgroup of	#72303941, #72304088, #72304090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4024842);