#20320

order	256
nilpotency class	3
isoclinism	isoclinic class of #20286
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	C₄²	C₂⁴
prime radical	#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#20320, #6361264, #6361267, …
maximal normal subgroup of	C₂×#20320, #6361264, #6361267, …
maximal quotient of	#220929, #220931, #220935, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13308);