#4397576

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=fhi,e^2=h,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, 4304772);