order	64
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #72225731 (order 4096) · Inn(G) = C₂³ · Out(G) = #845297
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Subgroups

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

161 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 14 points:

(1,2);(4,6)(7,8,10,12)(9,13,14,11);(7,9,10,14)(8,11,12,13);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 204);