#189260

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1001 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285685, #72285957, #72285969, …
maximal normal subgroup of	#72285685, #72285957, #72285969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96456);