#358837

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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)(7,17,20,26)(8,14)(9,15)(10,13)(11,16)(12,19,27,24)(18,31,32,35)(21,30)(22,29)(23,28)(25,33,36,34);(5,7)(6,12,16,27)(8,18)(9,19)(10,24)(11,20)(13,26,15,17)(14,25,30,36)(21,32)(22,33)(23,34)(28,35,29,31);(2,4)(6,13,16,15)(8,21)(12,17,27,26)(14,29,30,28)(18,32)(22,23)(25,35,36,31)(33,34);(5,8,9,22,11,21,10,23)(6,14,15,29,16,30,13,28)(7,18,19,33,20,32,24,34)(12,25,17,31,27,36,26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266033);