#116380

order	512
nilpotency class	3
isoclinism	isoclinic class of #116380
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233844, #72233853, #72233854
maximal normal subgroup of	#72233844, #72233853, #72233854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23576);