#3458

order	128
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^7|a^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=cf

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72233782
Fitting subgroup of	#3458⋊C₃, #11714230, #11715579, …
maximal subgroup of	#8410, #11389, #11391, …
maximal normal subgroup of	#8410, #11389, #11391, …
maximal quotient of	#34052, #34074, C₄×#543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2190);