#24860

order	256
nilpotency class	3
isoclinism	isoclinic class of #24398
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	C₂×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#24860, #6445663, #6445665, …
maximal normal subgroup of	C₂×#24860, #6445663, #6445665, …
maximal quotient of	#279248, #279254, #279260, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 17848);