#62840

order	256
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

3304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(2,6)(7,12)(8,13)(14,16)(17,18)(19,27)(20,26)(21,25)(22,23)(24,28)(29,31)(30,32);(4,11)(8,13)(9,15)(14,16)(18,23)(19,28)(25,26)(29,30);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20,22,21)(18,25,23,26)(19,29,28,30)(24,31,27,32);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,21,22,20)(18,26,23,25)(19,30,28,29)(24,32,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72272624
Fitting subgroup of	#62840⋊C₃
maximal subgroup of	#186161, #186168, #186175, …
maximal normal subgroup of	#186161, #186168, #186175, …
maximal quotient of	#7626238, #7626281, #7626285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55828);