#35920

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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

849 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 455 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,6,4)(3,8,5,7)(10,13)(11,15)(17,18,21,19)(20,23,24,22);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(1,2,6,4)(3,8,5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24);(1,4,6,2)(3,8,5,7);(1,5,6,3)(2,7,4,8)

Polycyclic presentation:

2^8|a^2=fg,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	#11709912, #11709913, #11709914
maximal subgroup of	#6579562, #6579566, #6579615, …
maximal normal subgroup of	#6579562, #6579566, #6579615, …
maximal quotient of	#517951, #517959, #517974, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 28908);