#376771

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14185
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 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:

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

Polycyclic presentation:

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

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, 283967);