#127709

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,d^b=dh

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, 34905);