#298244

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,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, 205440);