#106381

order	512
nilpotency class	3
isoclinism	isoclinic class of #106380
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211189
maximal normal subgroup of	#72211189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13577);