#17072

order	256
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

757 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=f,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#17072⋊C₃, #72227668, #72227672, …
Fitting subgroup of	#17072⋊C₃, #72227673, #72266666
maximal subgroup of	#104709, #104710, #104711, …
maximal normal subgroup of	#104709, #104710, #104711, …
maximal quotient of	#155847, #156395, #156411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10060);