#62908

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
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3339 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:

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

Polycyclic presentation:

2^8|b^2=gh,c^2=g,e^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

maximal subgroup of	#346807, #346831, #346855, …
maximal normal subgroup of	#346807, #346831, #346855, …
maximal quotient of	#7626586, #7626591, #7626669, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55896);