#270601

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1625	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177797);