#370155

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	#14043
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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)(3,6)(4,15,19,9)(5,14)(7,12)(8,11)(10,20)(13,17,24,18)(16,29,31,27)(21,26)(22,25)(23,30,28,32);(1,3,6,12,7,11,8,2)(4,15,17,24,19,9,18,13)(5,14,22,25,21,26,20,10)(16,28,32,29,31,23,30,27);(1,4,8,18,7,19,6,17)(2,9,3,13,12,15,11,24)(5,16,20,30,21,31,22,32)(10,23,14,27,25,28,26,29);(1,5)(2,10)(3,14)(4,16)(6,22)(7,21)(8,20)(9,23)(11,26)(12,25)(13,27)(15,28)(17,32)(18,30)(19,31)(24,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,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, 277351);