#25317

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	C₂²×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

345 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)(3,7)(4,5)(6,8)(9,10,14,18,13,17,27,34)(11,19,22,36,21,15,29,31)(12,23,26,33,25,35,28,16)(20,37,38,30,24,40,39,32);(2,5)(6,7)(9,11,14,22,13,21,27,29)(10,15,18,31,17,19,34,36)(12,24,26,39,25,20,28,38)(16,32,23,37,33,30,35,40);(9,12,13,25)(10,16,17,33)(11,20,21,24)(14,28,27,26)(15,30,19,32)(18,35,34,23)(22,39,29,38)(31,37,36,40);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,33)(18,34)(23,35)(30,32)(31,36)(37,40)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=h,d^2=eh,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×#25317, #6448073, #6448075, …
maximal normal subgroup of	C₂×#25317, #6448073, #6448075, …
maximal quotient of	#286656, #286662, #286668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18305);