#4396793

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1826 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fghi,d^2=gi,e^2=f,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, 4303989);