#120708

order	512
nilpotency class	3
isoclinism	isoclinic class of #117904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^c=gh

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, 27904);