C₂²×#137 #3579

order	128
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
prime radical	C₂²×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2311);