#30840

order	256
nilpotency class	3
isoclinism	isoclinic class of #30428
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

309 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6669695, #6669703, #6669709, …
maximal normal subgroup of	#6669695, #6669703, #6669709, …
maximal quotient of	#338684, #338685, #338688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 23828);