#105661

order	512
nilpotency class	3
isoclinism	isoclinic class of #105652
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

976 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239083, #72240726, #72240729
maximal normal subgroup of	#72239083, #72240726, #72240729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12857);