#369871

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14036
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1103 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)(3,9,16,27)(4,19,21,29)(5,10,22,28)(6,12)(7,13)(8,11)(14,24)(15,30)(17,25,18,26)(20,32)(23,31);(1,3)(2,9,11,27)(4,20)(5,15)(6,18)(7,17)(8,16)(10,24,28,30)(12,26,13,25)(14,22)(19,23,29,32)(21,31);(1,4,7,22,8,21,6,5)(2,10,13,19,11,28,12,29)(3,14,18,20,16,15,17,31)(9,23,26,24,27,32,25,30);(1,5,6,21,8,22,7,4)(2,10,13,19,11,28,12,29)(3,15,18,31,16,14,17,20)(9,24,25,23,27,30,26,32)

Polycyclic presentation:

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

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, 277067);