#376292

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312195, #72312212, #72312223
maximal normal subgroup of	#72312195, #72312212, #72312223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283488);