C₂×#11399 #479896

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈² · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₂×#11399

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₆	C₂²
commutator	C₂	C₂²×C₈²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(11,12,14,18,15,19,24,30,16,20,25,31,26,32,36,40)(13,17,22,28,23,29,35,39,21,27,33,37,34,38,41,42);(3,4,5,7,6,8,9,10)(11,13)(12,17)(14,22)(15,23)(16,21)(18,28)(19,29)(20,27)(24,35)(25,33)(26,34)(30,39)(31,37)(32,38)(36,41)(40,42);(13,21)(17,27)(22,33)(23,34)(28,37)(29,38)(35,41)(39,42)

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#129044, C₂×#128250, C₂×C₈×#504
maximal normal subgroup of	C₂×#129044, C₂×#128250, C₂×C₈×#504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387092);

C2×#11399 #479896