#606832

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514028);