C₂×#11416 #479965

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#11416

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 50
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,29,30,41)(22,33)(23,28)(24,26)(25,32,37,44)(27,35)(31,45,43,42)(34,48)(36,39)(38,50,47,46)(40,49);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,22,31,23,32,34,46,24,30,35,43,36,44,49,50)(20,25,27,38,28,29,40,42,26,37,33,47,39,41,48,45);(5,11)(8,14)(12,17)(15,18)(20,26)(25,37)(27,33)(28,39)(29,41)(38,47)(40,48)(42,45)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#483885, C₂×#483888, C₂×#479981, …
maximal normal subgroup of	C₂×#483885, C₂×#483888, C₂×#479981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387161);