#119525

order	512
nilpotency class	3
isoclinism	isoclinic class of #99899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

752 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26721);