#181784

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=h,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239269, #72239275, #72239287
maximal normal subgroup of	#72239269, #72239275, #72239287
maximal quotient of	#72239280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88980);