C₂²×#2197 #7622855

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
prime radical	C₂²×B(4,𝔽₂)	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4506 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530051);