#202928

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
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

748 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)(5,6,11,14)(7,13,18,20)(8,12,17,19)(9,16,10,15)(21,22)(23,27)(24,30)(25,29)(26,31)(28,32)(33,36)(34,35);(1,3,4,2)(5,7,10,17,11,18,9,8)(6,12,16,20,14,19,15,13)(21,23)(22,27)(24,34)(25,32)(26,33)(28,29)(30,35)(31,36);(5,8,9,18,11,17,10,7)(6,13,15,19,14,20,16,12)(21,23)(22,28)(24,32)(25,34)(26,33)(27,29)(30,36)(31,35);(6,14)(12,19)(13,20)(15,16)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,36,28)

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=dhi,f^c=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110124);