D₈×#124 #6579230

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×D₄×D₈	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2634 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

maximal subgroup of	#72282092, #72283243
maximal normal subgroup of	#72282092, #72283243

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486426);