#4416073

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1884 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)(9,12)(13,14,17,18)(15,21,23,19)(16,20,25,24)(22,27,28,26)(30,34)(31,37)(32,39)(42,44);(5,6,8,10)(7,9,11,12)(14,18)(19,21)(20,24)(26,27)(29,30)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(43,44);(1,2)(3,4)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,30)(31,35)(32,40)(33,34)(36,39)(37,41)(38,44)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,31,33,37)(30,35,34,41)(32,38,39,43)(36,42,40,44);(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,32,33,39)(30,36,34,40)(31,38,37,43)(35,42,41,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4323269);