#4987789

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1710 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)(7,11)(8,9)(10,12)(13,14,17,18)(15,19,22,26)(16,23,25,20)(21,28,24,27)(29,30)(31,36)(32,33)(34,35)(37,42)(38,43)(39,44)(40,41);(5,7,10,9)(6,11,12,8)(14,18)(19,26)(20,23)(27,28)(30,35)(31,38)(33,37)(39,41);(5,8)(6,7)(9,12)(10,11)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27)(29,31,34,38)(30,36,35,43)(32,41,42,39)(33,40,37,44);(1,2,3,4)(5,9,10,7)(6,11,12,8)(13,16)(14,20)(15,21)(17,25)(18,23)(19,27)(22,24)(26,28)(29,32)(30,37)(31,39)(33,35)(34,42)(36,40)(38,41)(43,44);(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,25)(24,28)(29,33,34,37)(30,32,35,42)(31,40,38,44)(36,41,43,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4894985);