#63092

order	256
nilpotency class	3
isoclinism	isoclinic class of #63091
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂⁴×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5398 conjugacy classes of subgroups; only the 3133 normal subgroups are shown.

normal subgroups only

This group has 5398 conjugacy classes of subgroups — too many to draw. Only its 3133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710899, #11710900, #63092⋊C₅, …
maximal subgroup of	#6573699, #6573702, #6573703, …
maximal normal subgroup of	#6573699, #6573702, #6573703, …
maximal quotient of	#10574063, #10574066, #10574096, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56080);