#60140

order	256
nilpotency class	2
isoclinism	isoclinic class of #534
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	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	#3449	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1361 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)(3,4)(5,6,9,14,10,13,20,26)(7,11,17,23,16,22,29,33)(8,12,18,25,19,24,30,34)(15,21,28,32,27,31,35,36);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,22)(14,23)(18,28)(19,27)(20,29)(24,31)(25,32)(26,33)(30,35)(34,36);(5,8,9,18,10,19,20,30)(6,12,14,25,13,24,26,34)(7,15,17,28,16,27,29,35)(11,21,23,32,22,31,33,36);(7,16)(11,22)(15,27)(17,29)(21,31)(23,33)(28,35)(32,36);(6,13)(11,22)(12,24)(14,26)(21,31)(23,33)(25,34)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#129498, #147530, #147531, …
maximal normal subgroup of	#129498, #147530, #147531, …
maximal quotient of	#6345185, #6345186, #6345187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53128);