#193957

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi

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, 101153);