#359314

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 266510);