#582924

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1900 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:

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

Polycyclic presentation:

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

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, 490120);