#25143

order	256
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25143, #6438879, #6438883, …
maximal normal subgroup of	C₂×#25143, #6438879, #6438883, …
maximal quotient of	#281536, #281537, #281540, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18131);