#4418854

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

1813 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)(5,6,9,12)(7,10,8,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,26)(22,27)(25,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,8)(6,10,12,11)(14,18)(19,21)(20,24)(27,28)(31,38)(35,37)(39,44)(42,43);(5,6)(7,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,32,34,30)(31,37,38,35)(33,40,41,36)(39,43,44,42);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fh,d^2=fg,e^2=f,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, 4326050);