#20481

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

481 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)(13,14,17,21,18,16,20,25)(15,19,24,27,23,22,26,28);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,22)(17,24)(18,23)(20,26)(21,27)(25,28);(13,16,17,25,18,14,20,21)(15,22,24,28,23,19,26,27);(5,7,8,11)(6,10,9,12)(14,16)(15,23)(21,25)(24,26)

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,b^a=be,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	#108668, #108739, #108743, …
maximal normal subgroup of	#108668, #108739, #108743, …
maximal quotient of	#183729, #183739, #183746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13469);