#36575

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 1179648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2622 conjugacy classes of subgroups; only the 595 normal subgroups are shown.

normal subgroups only

This group has 2622 conjugacy classes of subgroups — too many to draw. Only its 595 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) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 16 points:

(2,4)(6,8)(9,10,14,12)(11,15,13,16);(5,6)(7,8)(9,10,14,12)(11,16,13,15);(1,2)(3,4)(5,6)(7,8)(9,11,14,13)(10,15,12,16);(9,12,14,10)(11,15,13,16);(9,13,14,11)(10,16,12,15)

Polycyclic presentation:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710000, #11710001, #11710002, …
maximal subgroup of	#130707, #248461, #248557, …
maximal normal subgroup of	#130707, #248461, #248557, …
maximal quotient of	#518978, #518992, #519290, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29563);