Q₈×#376 #513476

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3566 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420672);