#116447

order	512
nilpotency class	3
isoclinism	isoclinic class of #98828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23643);