#4416560

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2002 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)(15,21)(16,23)(19,24)(20,25)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(14,18)(19,24)(20,25)(26,28)(29,31,34,32)(30,35,38,36)(33,39,42,40)(37,43,41,44);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,34,31)(30,36,38,35)(33,39,42,40)(37,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,35)(32,36)(33,37)(34,38)(39,43)(40,44)(41,42);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=h,e^2=f,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, 4323756);