#95365

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,e^2=hi,f^2=i,b^a=bd,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239640
maximal normal subgroup of	#72239640

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2561);