#4419082

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1927 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)(5,6,9,7)(8,11,12,10)(15,21)(16,23)(19,24)(20,25)(29,30)(31,34)(32,37)(33,38)(35,40)(36,42)(39,43)(41,44);(5,7,9,6)(8,11,12,10)(14,18)(19,24)(20,25)(26,28)(30,31)(35,37)(36,38)(43,44);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,30)(32,37,40,35)(33,38,42,36)(39,44,41,43);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,40)(30,35,31,37)(33,41,42,39)(36,44,38,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,31,38)(32,39,40,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=h,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, 4326278);