#210905

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
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

645 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)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22,26,30,27,31,25,35)(23,32,34,29,33,36,24,28);(5,7,8,15,10,17,9,16)(6,11,18,20,13,19,12,14)(21,23,26,34,27,33,25,24)(22,28,35,36,31,29,30,32);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19)(23,33)(24,34)(28,29)(32,36);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(21,24,25,33,27,34,26,23)(22,29,35,32,31,28,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118101);