#95057

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=ci,e^b=eh

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, 2253);