D₄×#564 #8092021

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

D₄×#564

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6091 conjugacy classes of subgroups; only the 3204 normal subgroups are shown.

normal subgroups only

This group has 6091 conjugacy classes of subgroups — too many to draw. Only its 3204 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4);(2,4);(7,13)(8,16)(10,14)(11,18)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(5,6,9,12)(7,14,13,10)(8,11,16,18)(15,20,19,17)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(5,7,9,13)(6,10,12,14)(8,15,16,19)(11,17,18,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8)(6,11)(7,15)(9,16)(10,17)(12,18)(13,19)(14,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999217);