Q₈×#392 #4307443

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 37748736
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

7814 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4214639);