#62904

order	256
nilpotency class	2
isoclinism	isoclinic class of #62892
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
prime radical	#3459	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3185 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#62904⋊C₃, #72268132, #72268152, …
maximal subgroup of	#346555, #346563, #346571, …
maximal normal subgroup of	#346555, #346563, #346571, …
maximal quotient of	#7626582, #7626666, #7626677, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55892);