#36066

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	Aut(G) order 1179648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

1881 conjugacy classes of subgroups; only the 569 normal subgroups are shown.

normal subgroups only

This group has 1881 conjugacy classes of subgroups — too many to draw. Only its 569 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) = 128

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

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

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=g,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	#11709938, #11709939, #11709940, …
maximal subgroup of	#132233, #248272, #248273, …
maximal normal subgroup of	#132233, #248272, #248273, …
maximal quotient of	#517984, #518000, #518007, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29054);