#31181

order	256
nilpotency class	3
isoclinism	isoclinic class of #30484
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	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

336 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)(3,5)(4,6)(7,8)(9,10,13,19,15,20,14,17)(11,16,23,31,24,29,21,32)(12,25,26,33,27,34,28,18)(22,36,35,39,38,40,37,30);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,22)(13,23)(14,21)(15,24)(17,32)(18,30)(19,31)(20,29)(25,36)(26,35)(27,38)(28,37)(33,39)(34,40);(1,4,5,8)(2,6,3,7)(10,17)(11,21)(13,14)(16,29)(18,25)(19,20)(22,35)(23,24)(26,28)(30,39)(33,34)(37,38);(2,3)(6,7)(9,12,15,27)(10,18,20,33)(11,22,24,38)(13,28,14,26)(16,30,29,39)(17,25,19,34)(21,35,23,37)(31,40,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6669699, #6669701, #6669703, …
maximal normal subgroup of	#6669699, #6669701, #6669703, …
maximal quotient of	#343953, #343954, #343957, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 24169);