#4381022

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,10,9)(7,12,11,8)(15,21)(16,23)(19,24)(20,25)(29,30)(31,38)(32,36)(33,35)(34,37)(39,44)(40,43)(41,42);(14,18)(19,24)(20,25)(26,28)(29,31,35,32)(30,36,33,38)(34,39,42,40)(37,43,41,44);(5,7)(6,8)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,36,33,38)(34,39,42,40)(37,44,41,43);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,30)(31,36,32,38)(34,41,42,37)(39,43,40,44);(5,9,10,6)(7,12,11,8)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,34,35,42)(30,37,33,41)(31,39,32,40)(36,43,38,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=fgh,e^2=fg,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, 4288218);