#4862093

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1828 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)(14,18)(15,21)(16,23)(26,28)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(22,27)(30,32)(31,38)(36,40)(39,44);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,30)(31,37,38,35)(33,36,41,40)(39,42,44,43);(5,7)(6,9)(8,11)(10,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4769289);