#270545

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×#3354	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,18)(17,20)(21,22)(23,27)(24,29)(25,28)(26,30)(31,36)(32,34)(33,35);(5,7,11,8)(6,12,15,20)(9,18,10,17)(13,19,16,14)(21,23,25,33)(22,27,28,35)(24,32,26,31)(29,34,30,36);(2,4)(5,8,11,7)(6,13,15,16)(9,18,10,17)(12,19,20,14)(22,28)(23,31)(24,26)(27,34)(32,33)(35,36);(5,9)(7,17)(8,18)(10,11)(12,20)(14,19)(21,24,25,26)(22,29,28,30)(23,32,33,31)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,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, 177741);