#5085418

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1856 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)(5,6,10,11)(7,9,8,12)(14,18)(15,21)(20,25)(23,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(5,7,10,8)(6,9,11,12)(16,24)(20,25)(23,27)(26,28)(30,34)(31,38)(36,40)(39,44);(6,11)(7,8)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,43)(41,44);(5,9)(6,7)(8,11)(10,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,d^2=ghi,e^2=gh,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, 4992614);