#28136

order	256
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,25)(13,18)(14,19)(15,20)(17,28)(21,34)(22,31)(23,32)(24,33)(26,39)(27,35)(29,36)(30,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11,14,23,15,24,13,22)(10,16,19,32,20,33,18,31)(12,26,27,38,29,37,28,21)(17,34,25,39,35,40,36,30);(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,30)(18,25)(19,36)(20,35)(22,26)(23,37)(24,38)(31,34)(32,40)(33,39);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,29,28)(16,31,33,32)(17,25,35,36)(21,26,38,37)(30,34,39,40)

Polycyclic presentation:

2^8|b^2=eg,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh,g^c=gh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#28136, #6459453, #6459460, …
maximal normal subgroup of	C₂×#28136, #6459453, #6459460, …
maximal quotient of	#316209, #316212, #316213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 21124);