#112933

order	512
nilpotency class	3
isoclinism	isoclinic class of #112929
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	#60558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

977 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)(3,9,12,6)(4,8)(5,7)(10,15,11,14)(13,16)(17,18,21,26,22,25,20,24)(19,27,30,23,29,32,28,31);(1,3)(2,6)(4,10)(5,12)(7,9)(8,14)(11,13)(15,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,31)(26,27);(3,10)(4,13)(6,14)(8,16)(9,15)(11,12)(23,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239675
maximal normal subgroup of	#72239675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20129);