#376294

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14176
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,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, 283490);