#183683

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

760 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)(7,13)(11,18,19,17)(15,20)(21,22,24,29,25,28,34,36)(23,30,32,35,31,27,33,26);(1,2,3,4)(5,6)(7,11)(8,16)(9,12)(10,14)(13,19)(15,18)(17,20)(21,23)(22,26)(24,33)(25,31)(27,29)(28,35)(30,36)(32,34);(5,7,14,20,9,13,8,15)(6,11,10,17,12,19,16,18)(21,23)(22,27)(24,32)(25,31)(26,29)(28,30)(33,34)(35,36);(22,28)(23,31)(29,36)(32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,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, 90879);