#25529

order	256
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

507 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)(12,23)(13,27)(14,18)(15,19)(17,26)(20,34)(21,32)(22,33)(24,39)(25,36)(28,31)(29,35)(30,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11,15,22)(10,16,19,33)(12,24,25,38)(13,21,14,28)(17,34,35,40)(18,31,27,32)(20,29,37,26)(23,39,36,30);(9,12,13,26,15,25,14,29)(10,17,18,23,19,35,27,36)(11,20,21,38,22,37,28,24)(16,30,31,34,33,39,32,40);(2,6)(5,7)(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25529, #6452225, #6452227, …
maximal normal subgroup of	C₂×#25529, #6452225, #6452227, …
maximal quotient of	#290897, #290898, #290899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18517);