#106444

order	512
nilpotency class	3
isoclinism	isoclinic class of #106434
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

945 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:

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

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=gi,d^b=dh,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	#72249767, #72249770, #72249779
maximal normal subgroup of	#72249767, #72249770, #72249779

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13640);