#7047

order	256
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#95
commutator	C₂²×C₄	C₄²
Frattini, cosocle	C₂⁴×C₄	C₂²
prime radical	C₂⁵	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,16,13)(12,14)(17,18,20,25,22,26,21,24)(19,27,28,23,30,31,29,32);(1,3)(2,5,6,7)(4,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,30,20,28)(18,23,25,31,26,32,24,27)

Polycyclic presentation:

2^8|a^2=d,b^2=e,c^2=fh,d^2=fh,e^2=fh,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#7047⋊C₃, #72266635, C₂×#7047⋊C₃
Fitting subgroup of	#7047⋊C₃
maximal subgroup of	#93824, #93834, C₂×#7047, …
maximal normal subgroup of	#93824, #93834, C₂×#7047, …
maximal quotient of	#92812, #93641, #93642, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35);