#298265

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205461);