D₄×#391 #519091

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5946 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:

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

maximal subgroup of	#72261354, #72261499, #72261784, …
maximal normal subgroup of	#72261354, #72261499, #72261784, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426287);