C₂²×#2396 #7627697

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

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

11058 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2);(3,4);(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(10,13)(14,16);(6,8)(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,14,13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534893);