C₂³×C₂²⋊Q₈ #62657

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₂³×C₂²⋊Q₈

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

7039 conjugacy classes of subgroups; only the 3695 normal subgroups are shown.

normal subgroups only

This group has 7039 conjugacy classes of subgroups — too many to draw. Only its 3695 normal subgroups are shown.

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

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

A minimal faithful permutation representation — 6 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55645);