#4416080

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=f,e^2=h,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, 4323276);