#284278

order	512
nilpotency class	3
isoclinism	isoclinic class of #264491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

707 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191474);