C₂×#24002 #6440296

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#121	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6718 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347492);