#118233

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

652 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fi,d^c=di

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, 25429);