C₂×#2427 #34697

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂²×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1947 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27685);