#113887

order	512
nilpotency class	3
isoclinism	isoclinic class of #113886
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1118 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)(3,9,11,6)(4,7)(5,8)(10,13,16,14)(12,15)(17,18)(19,27)(20,23)(21,25)(22,26)(24,28)(29,31)(30,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(18,23)(20,21)(24,27)(25,26)(28,30)(31,32);(1,4,5,12)(2,7,8,15)(3,10,11,16)(6,13,9,14)(17,19,22,29)(18,24,26,32)(20,28,21,30)(23,31,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257614, #72257699, #72257708
maximal normal subgroup of	#72257614, #72257699, #72257708
maximal quotient of	#72264382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21083);