#95005

order	512
nilpotency class	3
isoclinism	isoclinic class of #94966
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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

574 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:

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

Polycyclic presentation:

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

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, 2201);