#110956

order	512
nilpotency class	3
isoclinism	isoclinic class of #110956
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18152);