#5080299

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1689 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)(10,12)(13,14)(15,20)(16,24)(17,22)(18,19)(21,25)(23,27)(26,28)(30,31)(32,39)(36,38)(40,44);(6,9)(10,12)(14,19)(15,22)(16,25)(27,28)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,17)(16,26,25,23)(21,27,24,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(13,16)(14,21)(15,23)(17,27)(18,25)(19,24)(20,28)(22,26)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,20)(14,15,19,22)(16,27,25,28)(21,23,24,26)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=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, 4987495);