#4388653

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4295849);