#25265

order	256
nilpotency class	3
isoclinism	isoclinic class of #25140
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	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

364 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,16,23,32,24,33,22,31)(12,25,27,35,28,36,29,17)(21,37,38,34,26,40,39,30);(2,5)(6,8)(9,11,14,23,15,24,13,22)(10,16,19,32,20,33,18,31)(12,26,27,39,28,21,29,38)(17,34,25,40,35,30,36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,25)(19,36)(20,35)(22,38)(23,39)(24,26)(31,37)(32,40)(33,34);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,28,29)(16,31,33,32)(17,25,35,36)(21,38,26,39)(30,37,34,40)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=e,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×#25265, #6441696, #6441698, …
maximal normal subgroup of	C₂×#25265, #6441696, #6441698, …
maximal quotient of	#285628, #285629, #285640, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18253);