C₂×#9920 #234462

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2617	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2598 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

maximal subgroup of	C₂×#246012, C₂×#246103, C₂×#234601, …
maximal normal subgroup of	C₂×#246012, C₂×#246103, C₂×#234601, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141658);