#147188

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,17,12,19,32,16)(7,20,23,47,25,48,55,38)(8,26,29,43,31,42,28,14)(9,18,11,15,33,44,34,45)(13,24,37,56,39,57,36,22)(21,49,52,66,54,62,51,46)(27,40,30,59,61,65,60,41)(35,50,58,53,64,68,63,67);(2,4)(5,7,11,24,10,23,33,56,12,25,34,57,32,55,9,22)(6,13,18,38,16,36,45,48,19,39,44,47,17,37,15,20)(8,21,30,53,29,52,61,68,31,54,60,67,28,51,27,50)(14,35,41,62,42,63,65,66,43,64,59,49,26,58,40,46);(5,8,10,29,12,31,32,28)(6,14,17,26,19,43,16,42)(7,21,23,52,25,54,55,51)(9,27,11,30,33,61,34,60)(13,35,37,58,39,64,36,63)(15,40,44,59,45,65,18,41)(20,46,47,49,48,66,38,62)(22,50,24,53,56,68,57,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54384);