C₂×#34739 #7627903

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

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

9550 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535099);