#4315023

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1840 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)(29,30,34,35)(31,32,36,40)(33,41,42,37)(38,44,43,39);(5,6,8,10)(7,9,11,12)(14,18)(19,24)(20,25)(26,28)(30,35)(31,36)(33,42)(39,44);(5,7)(6,9)(8,11)(10,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,32)(33,38)(34,36)(35,40)(37,39)(41,44)(42,43);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,35)(33,39)(34,40)(37,43)(38,41)(42,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,37,35,41)(31,38,36,43)(32,39,40,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=fh,e^2=f,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, 4222219);