#555992

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1798 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 463188);