C₂³×#478 #10575012

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9874 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 6 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482208);