#494865

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 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:

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

Polycyclic presentation:

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

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, 402061);