#126905

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,d^b=di,e^b=egi

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, 34101);