#183684

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90880);