#146925

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286901, #72287546
maximal normal subgroup of	#72286901, #72287546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54121);