#192020

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

882 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)(6,10)(7,13)(8,15)(12,16)(14,17)(18,20)(21,22,25,30,27,24,26,29)(23,31,34,32,35,36,33,28);(2,4)(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,19)(17,20)(21,23,25,34,27,35,26,33)(22,28,30,31,24,32,29,36);(5,7,9,14)(6,11,12,19)(8,17,15,13)(10,18,16,20)(21,24,27,22)(23,32,35,28)(25,29,26,30)(31,34,36,33);(6,12)(10,16)(11,19)(18,20)(22,24)(28,32)(29,30)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99216);