#2845

order	128
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,15,16)(18,21)(22,23);(3,7)(5,8)(10,12)(14,16)(17,18,20,21)(19,22,24,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,21)(22,24);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,14,15,16)(17,19,20,24)(18,22,21,23)

Polycyclic presentation:

2^7|a^2=ef,b^2=g,c^2=eg,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#2845⋊C₃, #10360⋊C₃, #10586⋊C₃, …
Fitting subgroup of	#2845⋊C₃
maximal subgroup of	#10360, #10361, #10585, …
maximal normal subgroup of	#10360, #10361, #10585, …
maximal quotient of	#14640, #14677, #16525, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1577);