#4157594

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fg,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4064790);