#182553

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 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)(5,6,9,13)(7,15,16,20)(8,14,10,12)(11,18,19,17)(21,22,25,29,26,28,34,35)(23,27,31,24,30,32,36,33);(2,4)(5,7,10,18,9,16,8,17)(6,11,14,20,13,19,12,15)(21,23)(22,27)(24,29)(25,31)(26,30)(28,32)(33,35)(34,36);(21,24,25,32,26,33,34,27)(22,23,29,31,28,30,35,36);(7,16)(11,19)(15,20)(17,18)(22,28)(23,30)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89749);