#494897

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 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)(6,11)(7,16)(8,12)(9,20)(13,28)(14,23)(15,31)(17,19)(18,26)(21,27)(22,30)(24,25)(29,34)(32,33);(1,3,4,2)(5,6,12,25,9,15,21,23,10,14,27,31,20,24,8,11)(7,13,26,35,18,28,16,30,19,29,33,36,32,34,17,22);(5,7,8,17,20,32,27,33,10,19,21,16,9,18,12,26)(6,13,11,22,24,34,31,36,14,29,23,30,15,28,25,35);(6,14)(11,23)(13,29)(15,24)(22,30)(25,31)(28,34)(35,36)

Polycyclic presentation:

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

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, 402093);