#355851

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226669, #72226671, #72226674
maximal normal subgroup of	#72226669, #72226671, #72226674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263047);