#4315059

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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

2004 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)(15,21)(16,23)(19,24)(20,25)(29,30,34,31)(32,37,40,35)(33,38,41,36)(39,42,44,43);(5,6,9,11)(7,8,10,12)(14,18)(19,24)(20,25)(26,28)(30,31)(33,41)(35,37)(39,44);(5,7,9,10)(6,8,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,30)(32,35,40,37)(33,38,41,36)(39,42,44,43);(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)(33,39,41,44)(36,42,38,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,38)(32,39)(34,41)(35,42)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4222255);