#63002

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

5580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(5,6,10,11)(7,9,13,18)(8,12,17,19)(14,16,15,20);(1,2)(3,4);(6,11)(9,18)(12,19)(16,20);(5,7,10,13)(6,9,11,18)(8,15,17,14)(12,20,19,16);(5,8,10,17)(6,12,11,19)(7,14,13,15)(9,16,18,20);(5,9)(6,13)(7,11)(8,16)(10,18)(12,15)(14,19)(17,20)

Polycyclic presentation:

2^8|a^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63002⋊C₃, C₂²⋊(#585⋊C₅)
maximal subgroup of	#6573662, #6573668, #6573674, …
maximal normal subgroup of	#6573662, #6573668, #6573674, …
maximal quotient of	C₂²⋊C₄×2₋¹⁺⁴, #7625789, #7625790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55990);