#300560

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207756);