C₂³×#533 #10585898

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
socle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

39730 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493094);