#205396

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,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, 112592);