#60169

order	256
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator	C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=g,f^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dh

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#361317, #361319, #364102, …
maximal normal subgroup of	#361317, #361319, #364102, …
maximal quotient of	#6344526, #6344562, #6344566, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53157);