C₂×#9949 #235033

order	512
nilpotency class	3
isoclinism	isoclinic class of #9946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2396 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142229);