#284340

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

709 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)(21,22,25,29,26,30,24,28)(23,27,34,36,32,35,33,31);(1,2)(3,4)(6,10)(7,15)(8,17)(11,13)(12,20)(14,19)(21,23,24,33,26,32,25,34)(22,27,28,31,30,35,29,36);(2,4)(6,11)(10,13)(12,19)(14,20)(21,22,26,30)(23,31,32,36)(24,29,25,28)(27,34,35,33);(5,7,9,15)(6,12,11,19)(8,18,17,16)(10,14,13,20)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=i,d^2=h,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, 191536);