#494779

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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)(3,4)(5,6,10,17,11,15,28,33,12,14,29,32,9,16,27,34)(7,18,21,23,19,13,30,24,22,31,35,8,20,25,36,26);(5,7,10,21,11,19,28,30,12,22,29,35,9,20,27,36)(6,13,17,24,15,31,33,8,14,25,32,26,16,18,34,23);(2,4)(5,8,9,24,12,23,11,26)(6,7,15,19,14,22,16,20)(10,13,27,18,29,25,28,31)(17,35,33,36,32,21,34,30);(6,14)(8,23)(13,25)(15,16)(17,32)(18,31)(24,26)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=ch,e^b=ei,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, 401975);