#126718

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,30,40)(20,33)(21,36)(22,27)(23,29,38,31)(24,34)(25,41)(26,37)(32,47,44,43)(35,42)(39,46,48,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,31,21,32,34,46,22,30,35,29,37,44,41,45)(18,23,25,28,26,39,42,43,27,38,24,40,36,48,33,47);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch,d^b=di,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285512, #72285724, #72285735
maximal normal subgroup of	#72285512, #72285724, #72285735

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33914);