#210972

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

669 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)(7,14)(8,12)(9,19)(10,13)(11,16)(15,18)(17,20)(21,22,25,29,26,24,28,33)(23,27,32,35,30,31,34,36);(5,7,9,16,10,17,8,18)(6,11,12,14,13,15,19,20)(21,23)(22,27)(24,31)(25,32)(26,30)(28,34)(29,35)(33,36);(2,4)(5,6)(7,15)(8,19)(9,12)(10,13)(11,17)(14,18)(16,20)(23,30)(27,31)(32,34)(35,36);(5,8,10,9)(6,12,13,19)(7,16,17,18)(11,20,15,14)(21,24,25,33,26,22,28,29)(23,31,32,36,30,27,34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=dh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118168);