#4668475

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1725 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)(5,6,11,10)(7,8,12,9)(14,18)(15,21)(20,25)(22,27)(29,30,35,40)(31,37,32,36)(33,42,43,38)(34,41,44,39);(5,7)(6,9)(8,10)(11,12)(15,21)(19,24)(22,27)(26,28)(29,31,35,32)(30,36,40,37)(33,39,43,41)(34,42,44,38);(5,8,11,9)(6,12,10,7)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,35,31)(30,37,40,36)(33,39,43,41)(34,42,44,38);(1,2,3,4)(5,9,11,8)(6,12,10,7)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33)(30,38)(31,41)(32,39)(34,37)(35,43)(36,44)(40,42);(5,10,11,6)(7,8,12,9)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34)(30,39)(31,42)(32,38)(33,37)(35,44)(36,43)(40,41)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=gh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,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, 4575671);