#4121551

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1956 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)(3,5,7,8)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,39)(38,40);(11,15)(14,16)(17,19,22,20)(18,23,21,24)(26,28)(29,37)(31,33)(34,38);(9,10)(11,14)(12,13)(15,16)(17,18,22,21)(19,24,20,23)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,21,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,39,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,23,20,24)(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=h,c^2=h,d^2=fh,e^2=ghi,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	#72268481, #72268496
maximal normal subgroup of	#72268481, #72268496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4028747);