#202914

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

696 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=di,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110110);