#36641

order	256
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1065 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 1065 conjugacy classes of subgroups — too many to draw. Only its 487 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,6,7,8)(17,19,20,23)(18,21,22,24);(10,13)(14,16);(1,4,5,3)(2,6,7,8)(18,22)(21,24);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^8|b^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72237330
Fitting subgroup of	#36641⋊C₃
maximal subgroup of	#133995, #134204, #161360, …
maximal normal subgroup of	#133995, #134204, #161360, …
maximal quotient of	#607575, #607599, #607600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29629);