#183679

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
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

780 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90875);