#15155

order	256
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁴
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₄⋊(#399⋊C₃), #72227871, #72228018
maximal subgroup of	#104065, #104097, #104129, …
maximal normal subgroup of	#104065, #104097, #104129, …
maximal quotient of	#154174, #154391, #154405, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 8143);