#112877

order	512
nilpotency class	3
isoclinism	isoclinic class of #112875
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35102	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=eh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249734, #72257700, #72257703
maximal normal subgroup of	#72249734, #72257700, #72257703
maximal quotient of	#72257788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20073);