#497761

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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 ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 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)(5,6,10,17,11,18,25,32,12,14,26,33,9,16,8,15)(7,19,23,35,24,13,21,30,20,31,27,28,22,29,36,34);(2,3)(5,7)(6,13,14,29)(8,21)(9,22)(10,27)(11,24)(12,20)(15,28,32,35)(16,19,18,31)(17,34,33,30)(23,26)(25,36);(6,14)(7,20)(8,25)(10,26)(16,18)(22,24)(28,35)(30,34);(1,3,4,2)(5,8,9,26,12,25,11,10)(6,15,16,33,14,32,18,17)(7,21,22,23,20,36,24,27)(13,28,19,30,29,35,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404957);