#333527

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 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)(5,6,11,13,10,9,8,14)(7,15,16,12,18,20,17,19)(23,30)(27,35)(31,36)(32,34);(5,7,10,18)(6,12,9,19)(8,16,11,17)(13,15,14,20)(21,22,25,29)(23,31,30,36)(24,28,26,33)(27,34,35,32);(2,4)(5,8,10,11)(6,13,9,14)(7,16,18,17)(12,15,19,20)(21,23)(22,27)(24,32)(25,30)(26,34)(28,36)(29,35)(31,33);(2,4)(5,9,10,6)(7,15,18,20)(8,14,11,13)(12,16,19,17)(21,24,25,26)(22,28,29,33)(23,32,30,34)(27,36,35,31)

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gi,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, 240723);