#4416087

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

1785 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)(7,10)(8,12)(13,14,18,19)(15,17,21,26)(16,23,25,20)(22,28,27,24)(30,31)(33,41)(35,37)(39,44);(5,6,9,11)(7,8,10,12)(14,19)(15,21)(20,23)(22,27)(29,30,34,31)(32,35,40,37)(33,36,41,38)(39,42,44,43);(1,2,3,4)(5,7,9,10)(6,8,11,12)(13,15)(14,17)(16,22)(18,21)(19,26)(20,24)(23,28)(25,27)(29,31,34,30)(32,37,40,35)(33,36,41,38)(39,42,44,43);(5,8)(6,10)(7,11)(9,12)(13,16,18,25)(14,20,19,23)(15,22,21,27)(17,24,26,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(13,17)(14,21)(15,19)(16,24)(18,26)(20,27)(22,23)(25,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,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, 4323283);