#111965

order	512
nilpotency class	3
isoclinism	isoclinic class of #111959
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35301	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1023 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)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,21,23,22,26,20,25)(19,24,27,30,28,32,29,31);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(18,23)(20,21)(24,31)(25,26)(27,29)(30,32);(6,15)(9,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239972, #72257776, #72257777
maximal normal subgroup of	#72239972, #72257776, #72257777
maximal quotient of	#72239975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19161);