#116135

order	512
nilpotency class	3
isoclinism	isoclinic class of #116135
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23331);