#359167

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250732, #72252967, #72256876
maximal normal subgroup of	#72250732, #72252967, #72256876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266363);