#33443

order	256
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	Aut(G) order 2048
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×C₈×D₄
prime radical	C₂×#135	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^8|a^2=f,c^2=f,e^2=h,f^2=g,g^2=h,b^a=be,d^a=dh,e^a=eh,e^b=eh,d^c=dh

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128713, #128714, #128725, …
maximal normal subgroup of	#128713, #128714, #128725, …
maximal quotient of	#483491, D₈×M₃₂, #483526, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26431);