#20482

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂²×C₄²
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,b^a=be,c^a=ch,d^a=dh,d^b=dh,d^c=dh

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#108446, #108447, #108451, …
maximal normal subgroup of	#108446, #108447, #108451, …
maximal quotient of	#183730, #183738, #183747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13470);