#386732

order	512
nilpotency class	3
isoclinism	isoclinic class of #384777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2337
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14438
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1095 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)(3,12,16,27)(4,17,21,9)(5,10)(6,22)(7,11)(8,15,24,14)(13,25)(18,23)(19,31,20,26)(28,32)(29,30);(1,3)(2,8)(4,18)(5,14)(6,15)(7,16)(9,25)(10,27)(11,24)(12,22)(13,20)(17,30)(19,29)(21,28)(23,26)(31,32);(1,4)(2,9)(3,13,16,29)(5,19)(6,20)(7,21)(8,23,24,32)(10,26)(11,17)(12,25,27,30)(14,18,15,28)(22,31);(2,10,11,22)(3,14,16,15)(8,24)(9,26,17,31)(12,27)(13,18,29,28)(23,32)(25,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293928);