#4435206

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1764 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)(7,11)(10,12)(13,14)(15,17)(16,23)(18,19)(20,25)(21,27)(22,24)(26,28)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,6,9,8)(7,10,11,12)(14,19)(15,21)(20,23)(22,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,9,11)(6,10,8,12)(13,15,18,21)(14,17,19,27)(16,22,25,28)(20,26,23,24)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,8,9,6)(7,12,11,10)(13,16,18,25)(14,20,19,23)(15,22,21,28)(17,26,27,24)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(13,17)(14,21)(15,19)(16,24)(18,27)(20,22)(23,28)(25,26)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342402);