#36059

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	Aut(G) order 589824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

836 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 447 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) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11709933, #11709934, #11709935
maximal subgroup of	#6579570, #6579574, #6579621, …
maximal normal subgroup of	#6579570, #6579574, #6579621, …
maximal quotient of	#517955, #517959, #517973, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29047);