C₂×#29634 #6432944

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6774 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340140);