#20470

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

479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#108440, #108441, #108467, …
maximal normal subgroup of	#108440, #108441, #108467, …
maximal quotient of	#183823, #183824, #183825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13458);