#28278

order	256
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	Aut(G) = C₂¹¹×D₄ (order 16384) · Inn(G) = C₂²×D₄ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=eg,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^c=gh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#28278, #6465997, #6466007, …
maximal normal subgroup of	C₂×#28278, #6465997, #6466007, …
maximal quotient of	#318887, #318888, #318891, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 21266);