#112942

order	512
nilpotency class	3
isoclinism	isoclinic class of #112937
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	C₂×#3109	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238989, #72239075, #72240726, …
maximal normal subgroup of	#72238989, #72239075, #72240726, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20138);