D₄×#401 #4306521

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 44040192
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₄×#401

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2866 conjugacy classes of subgroups; only the 844 normal subgroups are shown.

normal subgroups only

This group has 2866 conjugacy classes of subgroups — too many to draw. Only its 844 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,3,4);(2,4);(7,11)(9,12)(14,17)(18,19)(21,22,24,26)(23,27,28,25);(5,6,8,10)(7,12,11,9)(13,14,16,17)(15,19,20,18)(21,22)(23,27)(24,26)(25,28);(5,7)(6,9)(8,11)(10,12)(13,15,16,20)(14,18,17,19)(21,23,24,28)(22,25,26,27)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4213717);