#284824

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192020);