#115093

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=di,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215258, #72216052, #72216055
maximal normal subgroup of	#72215258, #72216052, #72216055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22289);