D₄×#127 #36119

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2865 conjugacy classes of subgroups; only the 635 normal subgroups are shown.

normal subgroups only

This group has 2865 conjugacy classes of subgroups — too many to draw. Only its 635 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 12 points:

(10,12);(9,10)(11,12);(2,4)(6,7);(5,6,8,7);(1,2,3,4)(5,7,8,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29107);