#205412

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112608);