#19741

order	256
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^8|a^2=g,b^2=eg,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,d^c=dg

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19741, #6357385, #6357388, …
maximal normal subgroup of	C₂×#19741, #6357385, #6357388, …
maximal quotient of	#191528, #206278, #206280, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12729);