#62941

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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=g,e^2=h,f^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	#10580712, #10580713, #10580720, …
maximal normal subgroup of	#10580712, #10580713, #10580720, …
maximal quotient of	#7626612, #7626639, #7626645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55929);